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Resolución de integrales/ ((x^2-x+1)dx)/((x^4)+2(x^3)-3)

Integral de ((x^2-x+1)dx)/((x^4)+2(x^3)-3) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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0
∫01(x2−x)+1(x4+2x3)−3 dx\int\limits_{0}^{1} \frac{\left(x^{2} - x\right) + 1}{\left(x^{4} + 2 x^{3}\right) - 3}\, dx 
Integral((x^2 - x + 1)/(x^4 + 2*x^3 - 3), (x, 0, 1))
Solución detallada

  1. Hay varias maneras de calcular esta integral.

    Método #1

    1. Vuelva a escribir el integrando:

      (x2−x)+1(x4+2x3)−3=−x2−6x+710(x3+3x2+3x+3)+110(x−1)\frac{\left(x^{2} - x\right) + 1}{\left(x^{4} + 2 x^{3}\right) - 3} = - \frac{x^{2} - 6 x + 7}{10 \left(x^{3} + 3 x^{2} + 3 x + 3\right)} + \frac{1}{10 \left(x - 1\right)}

    2. Integramos término a término:

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫(−x2−6x+710(x3+3x2+3x+3)) dx=−∫x2−6x+7x3+3x2+3x+3 dx10\int \left(- \frac{x^{2} - 6 x + 7}{10 \left(x^{3} + 3 x^{2} + 3 x + 3\right)}\right)\, dx = - \frac{\int \frac{x^{2} - 6 x + 7}{x^{3} + 3 x^{2} + 3 x + 3}\, dx}{10}

        1. Vuelva a escribir el integrando:

          x2−6x+7x3+3x2+3x+3=x2x3+3x2+3x+3−6xx3+3x2+3x+3+7x3+3x2+3x+3\frac{x^{2} - 6 x + 7}{x^{3} + 3 x^{2} + 3 x + 3} = \frac{x^{2}}{x^{3} + 3 x^{2} + 3 x + 3} - \frac{6 x}{x^{3} + 3 x^{2} + 3 x + 3} + \frac{7}{x^{3} + 3 x^{2} + 3 x + 3}

        2. Integramos término a término:

          1. No puedo encontrar los pasos en la búsqueda de esta integral.

            Pero la integral

            RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫(−6xx3+3x2+3x+3) dx=−6∫xx3+3x2+3x+3 dx\int \left(- \frac{6 x}{x^{3} + 3 x^{2} + 3 x + 3}\right)\, dx = - 6 \int \frac{x}{x^{3} + 3 x^{2} + 3 x + 3}\, dx

            1. No puedo encontrar los pasos en la búsqueda de esta integral.

              Pero la integral

              RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))\operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}

            Por lo tanto, el resultado es: −6RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))- 6 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫7x3+3x2+3x+3 dx=7∫1x3+3x2+3x+3 dx\int \frac{7}{x^{3} + 3 x^{2} + 3 x + 3}\, dx = 7 \int \frac{1}{x^{3} + 3 x^{2} + 3 x + 3}\, dx

            1. No puedo encontrar los pasos en la búsqueda de esta integral.

              Pero la integral

              23log⁡(x+1+23)6−23log⁡(x2+x(2−23)−23+1+223)12+233atan⁡(2233x3−33+22333)6\frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{6} - \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{12} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{6}

            Por lo tanto, el resultado es: 723log⁡(x+1+23)6−723log⁡(x2+x(2−23)−23+1+223)12+7233atan⁡(2233x3−33+22333)6\frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{6} - \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{12} + \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{6}

          El resultado es: 723log⁡(x+1+23)6−723log⁡(x2+x(2−23)−23+1+223)12+7233atan⁡(2233x3−33+22333)6−6RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))+RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))\frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{6} - \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{12} + \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{6} - 6 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)} + \operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}

        Por lo tanto, el resultado es: −723log⁡(x+1+23)60+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(2233x3−33+22333)60+3RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10- \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{60} + \frac{3 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫110(x−1) dx=∫1x−1 dx10\int \frac{1}{10 \left(x - 1\right)}\, dx = \frac{\int \frac{1}{x - 1}\, dx}{10}

        1. que u=x−1u = x - 1.

          Luego que du=dxdu = dx y ponemos dudu:

          ∫1u du\int \frac{1}{u}\, du

          1. Integral 1u\frac{1}{u} es log⁡(u)\log{\left(u \right)}.

          Si ahora sustituir uu más en:

          log⁡(x−1)\log{\left(x - 1 \right)}

        Por lo tanto, el resultado es: log⁡(x−1)10\frac{\log{\left(x - 1 \right)}}{10}

      El resultado es: log⁡(x−1)10−723log⁡(x+1+23)60+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(2233x3−33+22333)60+3RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10\frac{\log{\left(x - 1 \right)}}{10} - \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{60} + \frac{3 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

    Método #2

    1. Vuelva a escribir el integrando:

      (x2−x)+1(x4+2x3)−3=x2(x4+2x3)−3−x(x4+2x3)−3+1(x4+2x3)−3\frac{\left(x^{2} - x\right) + 1}{\left(x^{4} + 2 x^{3}\right) - 3} = \frac{x^{2}}{\left(x^{4} + 2 x^{3}\right) - 3} - \frac{x}{\left(x^{4} + 2 x^{3}\right) - 3} + \frac{1}{\left(x^{4} + 2 x^{3}\right) - 3}

    2. Integramos término a término:

      1. Vuelva a escribir el integrando:

        x2(x4+2x3)−3=−x2−6x−310(x3+3x2+3x+3)+110(x−1)\frac{x^{2}}{\left(x^{4} + 2 x^{3}\right) - 3} = - \frac{x^{2} - 6 x - 3}{10 \left(x^{3} + 3 x^{2} + 3 x + 3\right)} + \frac{1}{10 \left(x - 1\right)}

      2. Integramos término a término:

        1. La integral del producto de una función por una constante es la constante por la integral de esta función:

          ∫(−x2−6x−310(x3+3x2+3x+3)) dx=−∫x2−6x−3x3+3x2+3x+3 dx10\int \left(- \frac{x^{2} - 6 x - 3}{10 \left(x^{3} + 3 x^{2} + 3 x + 3\right)}\right)\, dx = - \frac{\int \frac{x^{2} - 6 x - 3}{x^{3} + 3 x^{2} + 3 x + 3}\, dx}{10}

          1. Vuelva a escribir el integrando:

            x2−6x−3x3+3x2+3x+3=x2x3+3x2+3x+3−6xx3+3x2+3x+3−3x3+3x2+3x+3\frac{x^{2} - 6 x - 3}{x^{3} + 3 x^{2} + 3 x + 3} = \frac{x^{2}}{x^{3} + 3 x^{2} + 3 x + 3} - \frac{6 x}{x^{3} + 3 x^{2} + 3 x + 3} - \frac{3}{x^{3} + 3 x^{2} + 3 x + 3}

          2. Integramos término a término:

            1. No puedo encontrar los pasos en la búsqueda de esta integral.

              Pero la integral

              RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}

            1. La integral del producto de una función por una constante es la constante por la integral de esta función:

              ∫(−6xx3+3x2+3x+3) dx=−6∫xx3+3x2+3x+3 dx\int \left(- \frac{6 x}{x^{3} + 3 x^{2} + 3 x + 3}\right)\, dx = - 6 \int \frac{x}{x^{3} + 3 x^{2} + 3 x + 3}\, dx

              1. No puedo encontrar los pasos en la búsqueda de esta integral.

                Pero la integral

                RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))\operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}

              Por lo tanto, el resultado es: −6RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))- 6 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}

            1. La integral del producto de una función por una constante es la constante por la integral de esta función:

              ∫(−3x3+3x2+3x+3) dx=−3∫1x3+3x2+3x+3 dx\int \left(- \frac{3}{x^{3} + 3 x^{2} + 3 x + 3}\right)\, dx = - 3 \int \frac{1}{x^{3} + 3 x^{2} + 3 x + 3}\, dx

              1. No puedo encontrar los pasos en la búsqueda de esta integral.

                Pero la integral

                23log⁡(x+1+23)6−23log⁡(x2+x(2−23)−23+1+223)12+233atan⁡(2233x3−33+22333)6\frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{6} - \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{12} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{6}

              Por lo tanto, el resultado es: −23log⁡(x+1+23)2+23log⁡(x2+x(2−23)−23+1+223)4−233atan⁡(2233x3−33+22333)2- \frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{2} + \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{4} - \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{2}

            El resultado es: −23log⁡(x+1+23)2+23log⁡(x2+x(2−23)−23+1+223)4−233atan⁡(2233x3−33+22333)2−6RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))+RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))- \frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{2} + \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{4} - \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{2} - 6 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)} + \operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}

          Por lo tanto, el resultado es: 23log⁡(x+1+23)20−23log⁡(x2+x(2−23)−23+1+223)40+233atan⁡(2233x3−33+22333)20+3RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10\frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{20} - \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{40} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{20} + \frac{3 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

        1. La integral del producto de una función por una constante es la constante por la integral de esta función:

          ∫110(x−1) dx=∫1x−1 dx10\int \frac{1}{10 \left(x - 1\right)}\, dx = \frac{\int \frac{1}{x - 1}\, dx}{10}

          1. que u=x−1u = x - 1.

            Luego que du=dxdu = dx y ponemos dudu:

            ∫1u du\int \frac{1}{u}\, du

            1. Integral 1u\frac{1}{u} es log⁡(u)\log{\left(u \right)}.

            Si ahora sustituir uu más en:

            log⁡(x−1)\log{\left(x - 1 \right)}

          Por lo tanto, el resultado es: log⁡(x−1)10\frac{\log{\left(x - 1 \right)}}{10}

        El resultado es: log⁡(x−1)10+23log⁡(x+1+23)20−23log⁡(x2+x(2−23)−23+1+223)40+233atan⁡(2233x3−33+22333)20+3RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10\frac{\log{\left(x - 1 \right)}}{10} + \frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{20} - \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{40} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{20} + \frac{3 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

      1. La integral del producto de una función por una constante es la constante por la integral de esta función:

        ∫(−x(x4+2x3)−3) dx=−∫x(x4+2x3)−3 dx\int \left(- \frac{x}{\left(x^{4} + 2 x^{3}\right) - 3}\right)\, dx = - \int \frac{x}{\left(x^{4} + 2 x^{3}\right) - 3}\, dx

        1. Vuelva a escribir el integrando:

          x(x4+2x3)−3=−x2+4x−310(x3+3x2+3x+3)+110(x−1)\frac{x}{\left(x^{4} + 2 x^{3}\right) - 3} = - \frac{x^{2} + 4 x - 3}{10 \left(x^{3} + 3 x^{2} + 3 x + 3\right)} + \frac{1}{10 \left(x - 1\right)}

        2. Integramos término a término:

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫(−x2+4x−310(x3+3x2+3x+3)) dx=−∫x2+4x−3x3+3x2+3x+3 dx10\int \left(- \frac{x^{2} + 4 x - 3}{10 \left(x^{3} + 3 x^{2} + 3 x + 3\right)}\right)\, dx = - \frac{\int \frac{x^{2} + 4 x - 3}{x^{3} + 3 x^{2} + 3 x + 3}\, dx}{10}

            1. Vuelva a escribir el integrando:

              x2+4x−3x3+3x2+3x+3=x2x3+3x2+3x+3+4xx3+3x2+3x+3−3x3+3x2+3x+3\frac{x^{2} + 4 x - 3}{x^{3} + 3 x^{2} + 3 x + 3} = \frac{x^{2}}{x^{3} + 3 x^{2} + 3 x + 3} + \frac{4 x}{x^{3} + 3 x^{2} + 3 x + 3} - \frac{3}{x^{3} + 3 x^{2} + 3 x + 3}

            2. Integramos término a término:

              1. No puedo encontrar los pasos en la búsqueda de esta integral.

                Pero la integral

                RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}

              1. La integral del producto de una función por una constante es la constante por la integral de esta función:

                ∫4xx3+3x2+3x+3 dx=4∫xx3+3x2+3x+3 dx\int \frac{4 x}{x^{3} + 3 x^{2} + 3 x + 3}\, dx = 4 \int \frac{x}{x^{3} + 3 x^{2} + 3 x + 3}\, dx

                1. No puedo encontrar los pasos en la búsqueda de esta integral.

                  Pero la integral

                  RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))\operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}

                Por lo tanto, el resultado es: 4RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))4 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}

              1. La integral del producto de una función por una constante es la constante por la integral de esta función:

                ∫(−3x3+3x2+3x+3) dx=−3∫1x3+3x2+3x+3 dx\int \left(- \frac{3}{x^{3} + 3 x^{2} + 3 x + 3}\right)\, dx = - 3 \int \frac{1}{x^{3} + 3 x^{2} + 3 x + 3}\, dx

                1. No puedo encontrar los pasos en la búsqueda de esta integral.

                  Pero la integral

                  23log⁡(x+1+23)6−23log⁡(x2+x(2−23)−23+1+223)12+233atan⁡(2233x3−33+22333)6\frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{6} - \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{12} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{6}

                Por lo tanto, el resultado es: −23log⁡(x+1+23)2+23log⁡(x2+x(2−23)−23+1+223)4−233atan⁡(2233x3−33+22333)2- \frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{2} + \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{4} - \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{2}

              El resultado es: −23log⁡(x+1+23)2+23log⁡(x2+x(2−23)−23+1+223)4−233atan⁡(2233x3−33+22333)2+4RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))+RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))- \frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{2} + \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{4} - \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{2} + 4 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)} + \operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}

            Por lo tanto, el resultado es: 23log⁡(x+1+23)20−23log⁡(x2+x(2−23)−23+1+223)40+233atan⁡(2233x3−33+22333)20−2RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10\frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{20} - \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{40} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{20} - \frac{2 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

          1. La integral del producto de una función por una constante es la constante por la integral de esta función:

            ∫110(x−1) dx=∫1x−1 dx10\int \frac{1}{10 \left(x - 1\right)}\, dx = \frac{\int \frac{1}{x - 1}\, dx}{10}

            1. que u=x−1u = x - 1.

              Luego que du=dxdu = dx y ponemos dudu:

              ∫1u du\int \frac{1}{u}\, du

              1. Integral 1u\frac{1}{u} es log⁡(u)\log{\left(u \right)}.

              Si ahora sustituir uu más en:

              log⁡(x−1)\log{\left(x - 1 \right)}

            Por lo tanto, el resultado es: log⁡(x−1)10\frac{\log{\left(x - 1 \right)}}{10}

          El resultado es: log⁡(x−1)10+23log⁡(x+1+23)20−23log⁡(x2+x(2−23)−23+1+223)40+233atan⁡(2233x3−33+22333)20−2RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10\frac{\log{\left(x - 1 \right)}}{10} + \frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{20} - \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{40} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{20} - \frac{2 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

        Por lo tanto, el resultado es: −log⁡(x−1)10−23log⁡(x+1+23)20+23log⁡(x2+x(2−23)−23+1+223)40−233atan⁡(2233x3−33+22333)20+2RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5+RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10- \frac{\log{\left(x - 1 \right)}}{10} - \frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{20} + \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{40} - \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{20} + \frac{2 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} + \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

      1. Vuelva a escribir el integrando:

        1(x4+2x3)−3=−x2+4x+710(x3+3x2+3x+3)+110(x−1)\frac{1}{\left(x^{4} + 2 x^{3}\right) - 3} = - \frac{x^{2} + 4 x + 7}{10 \left(x^{3} + 3 x^{2} + 3 x + 3\right)} + \frac{1}{10 \left(x - 1\right)}

      2. Integramos término a término:

        1. La integral del producto de una función por una constante es la constante por la integral de esta función:

          ∫(−x2+4x+710(x3+3x2+3x+3)) dx=−∫x2+4x+7x3+3x2+3x+3 dx10\int \left(- \frac{x^{2} + 4 x + 7}{10 \left(x^{3} + 3 x^{2} + 3 x + 3\right)}\right)\, dx = - \frac{\int \frac{x^{2} + 4 x + 7}{x^{3} + 3 x^{2} + 3 x + 3}\, dx}{10}

          1. Vuelva a escribir el integrando:

            x2+4x+7x3+3x2+3x+3=x2x3+3x2+3x+3+4xx3+3x2+3x+3+7x3+3x2+3x+3\frac{x^{2} + 4 x + 7}{x^{3} + 3 x^{2} + 3 x + 3} = \frac{x^{2}}{x^{3} + 3 x^{2} + 3 x + 3} + \frac{4 x}{x^{3} + 3 x^{2} + 3 x + 3} + \frac{7}{x^{3} + 3 x^{2} + 3 x + 3}

          2. Integramos término a término:

            1. No puedo encontrar los pasos en la búsqueda de esta integral.

              Pero la integral

              RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}

            1. La integral del producto de una función por una constante es la constante por la integral de esta función:

              ∫4xx3+3x2+3x+3 dx=4∫xx3+3x2+3x+3 dx\int \frac{4 x}{x^{3} + 3 x^{2} + 3 x + 3}\, dx = 4 \int \frac{x}{x^{3} + 3 x^{2} + 3 x + 3}\, dx

              1. No puedo encontrar los pasos en la búsqueda de esta integral.

                Pero la integral

                RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))\operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}

              Por lo tanto, el resultado es: 4RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))4 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}

            1. La integral del producto de una función por una constante es la constante por la integral de esta función:

              ∫7x3+3x2+3x+3 dx=7∫1x3+3x2+3x+3 dx\int \frac{7}{x^{3} + 3 x^{2} + 3 x + 3}\, dx = 7 \int \frac{1}{x^{3} + 3 x^{2} + 3 x + 3}\, dx

              1. No puedo encontrar los pasos en la búsqueda de esta integral.

                Pero la integral

                23log⁡(x+1+23)6−23log⁡(x2+x(2−23)−23+1+223)12+233atan⁡(2233x3−33+22333)6\frac{\sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{6} - \frac{\sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{12} + \frac{\sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{6}

              Por lo tanto, el resultado es: 723log⁡(x+1+23)6−723log⁡(x2+x(2−23)−23+1+223)12+7233atan⁡(2233x3−33+22333)6\frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{6} - \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{12} + \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{6}

            El resultado es: 723log⁡(x+1+23)6−723log⁡(x2+x(2−23)−23+1+223)12+7233atan⁡(2233x3−33+22333)6+4RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))+RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))\frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{6} - \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{12} + \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{6} + 4 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)} + \operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}

          Por lo tanto, el resultado es: −723log⁡(x+1+23)60+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(2233x3−33+22333)60−2RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10- \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{60} - \frac{2 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

        1. La integral del producto de una función por una constante es la constante por la integral de esta función:

          ∫110(x−1) dx=∫1x−1 dx10\int \frac{1}{10 \left(x - 1\right)}\, dx = \frac{\int \frac{1}{x - 1}\, dx}{10}

          1. que u=x−1u = x - 1.

            Luego que du=dxdu = dx y ponemos dudu:

            ∫1u du\int \frac{1}{u}\, du

            1. Integral 1u\frac{1}{u} es log⁡(u)\log{\left(u \right)}.

            Si ahora sustituir uu más en:

            log⁡(x−1)\log{\left(x - 1 \right)}

          Por lo tanto, el resultado es: log⁡(x−1)10\frac{\log{\left(x - 1 \right)}}{10}

        El resultado es: log⁡(x−1)10−723log⁡(x+1+23)60+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(2233x3−33+22333)60−2RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10\frac{\log{\left(x - 1 \right)}}{10} - \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{60} - \frac{2 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

      El resultado es: log⁡(x−1)10−723log⁡(x+1+23)60+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(2233x3−33+22333)60+3RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10\frac{\log{\left(x - 1 \right)}}{10} - \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{60} + \frac{3 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10}

  2. Ahora simplificar:

    log⁡(x−1)10−723log⁡(x+1+23)60+3(−2236−236)log⁡(x−3−223−23+36(−2236−236)2)5+3(−236(−12−3i2)−223(−12−3i2)6)log⁡(x−3−23−12−3i2+36(−236(−12−3i2)−223(−12−3i2)6)2−223(−12−3i2))5+3(−223(−12+3i2)6−236(−12+3i2))log⁡(x−3−223(−12+3i2)+36(−223(−12+3i2)6−236(−12+3i2))2−23−12+3i2)5−(13+223(−12−3i2)3+236(−12−3i2))log⁡(x−35−3235(−12−3i2)+24(13+223(−12−3i2)3+236(−12−3i2))25−6⋅223(−12−3i2)5)10−(13+236(−12+3i2)+223(−12+3i2)3)log⁡(x−35−6⋅223(−12+3i2)5+24(13+236(−12+3i2)+223(−12+3i2)3)25−3235(−12+3i2))10−(236+13+2233)log⁡(x−6⋅2235−3235−35+24(236+13+2233)25)10+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(3(223x−1+223)3)60\frac{\log{\left(x - 1 \right)}}{10} - \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{3 \left(- \frac{2^{\frac{2}{3}}}{6} - \frac{\sqrt[3]{2}}{6}\right) \log{\left(x - 3 - 2^{\frac{2}{3}} - \sqrt[3]{2} + 36 \left(- \frac{2^{\frac{2}{3}}}{6} - \frac{\sqrt[3]{2}}{6}\right)^{2} \right)}}{5} + \frac{3 \left(- \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{6}\right) \log{\left(x - 3 - \frac{\sqrt[3]{2}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}} + 36 \left(- \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{6}\right)^{2} - 2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \right)}}{5} + \frac{3 \left(- \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{6} - \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}\right) \log{\left(x - 3 - 2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) + 36 \left(- \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{6} - \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}\right)^{2} - \frac{\sqrt[3]{2}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}} \right)}}{5} - \frac{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}\right) \log{\left(x - \frac{3}{5} - \frac{3 \sqrt[3]{2}}{5 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} + \frac{24 \left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}\right)^{2}}{5} - \frac{6 \cdot 2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{5} \right)}}{10} - \frac{\left(\frac{1}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3}\right) \log{\left(x - \frac{3}{5} - \frac{6 \cdot 2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{5} + \frac{24 \left(\frac{1}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3}\right)^{2}}{5} - \frac{3 \sqrt[3]{2}}{5 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} \right)}}{10} - \frac{\left(\frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3}\right) \log{\left(x - \frac{6 \cdot 2^{\frac{2}{3}}}{5} - \frac{3 \sqrt[3]{2}}{5} - \frac{3}{5} + \frac{24 \left(\frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3}\right)^{2}}{5} \right)}}{10} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2^{\frac{2}{3}} x - 1 + 2^{\frac{2}{3}}\right)}{3} \right)}}{60}

  3. Añadimos la constante de integración:

    log⁡(x−1)10−723log⁡(x+1+23)60+3(−2236−236)log⁡(x−3−223−23+36(−2236−236)2)5+3(−236(−12−3i2)−223(−12−3i2)6)log⁡(x−3−23−12−3i2+36(−236(−12−3i2)−223(−12−3i2)6)2−223(−12−3i2))5+3(−223(−12+3i2)6−236(−12+3i2))log⁡(x−3−223(−12+3i2)+36(−223(−12+3i2)6−236(−12+3i2))2−23−12+3i2)5−(13+223(−12−3i2)3+236(−12−3i2))log⁡(x−35−3235(−12−3i2)+24(13+223(−12−3i2)3+236(−12−3i2))25−6⋅223(−12−3i2)5)10−(13+236(−12+3i2)+223(−12+3i2)3)log⁡(x−35−6⋅223(−12+3i2)5+24(13+236(−12+3i2)+223(−12+3i2)3)25−3235(−12+3i2))10−(236+13+2233)log⁡(x−6⋅2235−3235−35+24(236+13+2233)25)10+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(3(223x−1+223)3)60+constant\frac{\log{\left(x - 1 \right)}}{10} - \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{3 \left(- \frac{2^{\frac{2}{3}}}{6} - \frac{\sqrt[3]{2}}{6}\right) \log{\left(x - 3 - 2^{\frac{2}{3}} - \sqrt[3]{2} + 36 \left(- \frac{2^{\frac{2}{3}}}{6} - \frac{\sqrt[3]{2}}{6}\right)^{2} \right)}}{5} + \frac{3 \left(- \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{6}\right) \log{\left(x - 3 - \frac{\sqrt[3]{2}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}} + 36 \left(- \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{6}\right)^{2} - 2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \right)}}{5} + \frac{3 \left(- \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{6} - \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}\right) \log{\left(x - 3 - 2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) + 36 \left(- \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{6} - \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}\right)^{2} - \frac{\sqrt[3]{2}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}} \right)}}{5} - \frac{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}\right) \log{\left(x - \frac{3}{5} - \frac{3 \sqrt[3]{2}}{5 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} + \frac{24 \left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}\right)^{2}}{5} - \frac{6 \cdot 2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{5} \right)}}{10} - \frac{\left(\frac{1}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3}\right) \log{\left(x - \frac{3}{5} - \frac{6 \cdot 2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{5} + \frac{24 \left(\frac{1}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3}\right)^{2}}{5} - \frac{3 \sqrt[3]{2}}{5 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} \right)}}{10} - \frac{\left(\frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3}\right) \log{\left(x - \frac{6 \cdot 2^{\frac{2}{3}}}{5} - \frac{3 \sqrt[3]{2}}{5} - \frac{3}{5} + \frac{24 \left(\frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3}\right)^{2}}{5} \right)}}{10} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2^{\frac{2}{3}} x - 1 + 2^{\frac{2}{3}}\right)}{3} \right)}}{60}+ \mathrm{constant}

Respuesta:

log⁡(x−1)10−723log⁡(x+1+23)60+3(−2236−236)log⁡(x−3−223−23+36(−2236−236)2)5+3(−236(−12−3i2)−223(−12−3i2)6)log⁡(x−3−23−12−3i2+36(−236(−12−3i2)−223(−12−3i2)6)2−223(−12−3i2))5+3(−223(−12+3i2)6−236(−12+3i2))log⁡(x−3−223(−12+3i2)+36(−223(−12+3i2)6−236(−12+3i2))2−23−12+3i2)5−(13+223(−12−3i2)3+236(−12−3i2))log⁡(x−35−3235(−12−3i2)+24(13+223(−12−3i2)3+236(−12−3i2))25−6⋅223(−12−3i2)5)10−(13+236(−12+3i2)+223(−12+3i2)3)log⁡(x−35−6⋅223(−12+3i2)5+24(13+236(−12+3i2)+223(−12+3i2)3)25−3235(−12+3i2))10−(236+13+2233)log⁡(x−6⋅2235−3235−35+24(236+13+2233)25)10+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(3(223x−1+223)3)60+constant\frac{\log{\left(x - 1 \right)}}{10} - \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{3 \left(- \frac{2^{\frac{2}{3}}}{6} - \frac{\sqrt[3]{2}}{6}\right) \log{\left(x - 3 - 2^{\frac{2}{3}} - \sqrt[3]{2} + 36 \left(- \frac{2^{\frac{2}{3}}}{6} - \frac{\sqrt[3]{2}}{6}\right)^{2} \right)}}{5} + \frac{3 \left(- \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{6}\right) \log{\left(x - 3 - \frac{\sqrt[3]{2}}{- \frac{1}{2} - \frac{\sqrt{3} i}{2}} + 36 \left(- \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} - \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{6}\right)^{2} - 2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \right)}}{5} + \frac{3 \left(- \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{6} - \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}\right) \log{\left(x - 3 - 2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) + 36 \left(- \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{6} - \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}\right)^{2} - \frac{\sqrt[3]{2}}{- \frac{1}{2} + \frac{\sqrt{3} i}{2}} \right)}}{5} - \frac{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}\right) \log{\left(x - \frac{3}{5} - \frac{3 \sqrt[3]{2}}{5 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} + \frac{24 \left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}\right)^{2}}{5} - \frac{6 \cdot 2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{5} \right)}}{10} - \frac{\left(\frac{1}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3}\right) \log{\left(x - \frac{3}{5} - \frac{6 \cdot 2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{5} + \frac{24 \left(\frac{1}{3} + \frac{\sqrt[3]{2}}{6 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3}\right)^{2}}{5} - \frac{3 \sqrt[3]{2}}{5 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} \right)}}{10} - \frac{\left(\frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3}\right) \log{\left(x - \frac{6 \cdot 2^{\frac{2}{3}}}{5} - \frac{3 \sqrt[3]{2}}{5} - \frac{3}{5} + \frac{24 \left(\frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3}\right)^{2}}{5} \right)}}{10} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2^{\frac{2}{3}} x - 1 + 2^{\frac{2}{3}}\right)}{3} \right)}}{60}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                              /                             /                   2\\                                                                                                                                                                                    /    ___    2/3   ___      2/3   ___\
 |                               |    3       2                |3       18*t   24*t ||                                                                                                                                                                    3 ___   ___     |  \/ 3    2   *\/ 3    x*2   *\/ 3 |
 |    2                   RootSum|12*t  - 12*t  - 1, t -> t*log|- + x - ---- + -----||                          /    3                      /                   2\\     3 ___    /        3 ___\     3 ___    /     2/3    2   3 ___     /    3 ___\\   7*\/ 2 *\/ 3 *atan|- ----- + ---------- + ------------|
 |   x  - x + 1                  \                             \5        5       5  //   log(-1 + x)   3*RootSum\36*t  - 6*t + 1, t -> t*log\-3 + x + 6*t + 36*t //   7*\/ 2 *log\1 + x + \/ 2 /   7*\/ 2 *log\1 + 2    + x  - \/ 2  + x*\2 - \/ 2 //                     \    3         3             3      /
 | ------------- dx = C - ------------------------------------------------------------ + ----------- + ------------------------------------------------------------ - -------------------------- + -------------------------------------------------- - -------------------------------------------------------
 |  4      3                                           10                                     10                                    5                                             60                                      120                                                      60                          
 | x  + 2*x  - 3                                                                                                                                                                                                                                                                                               
 |                                                                                                                                                                                                                                                                                                             
/
∫(x2−x)+1(x4+2x3)−3 dx=C+log⁡(x−1)10−723log⁡(x+1+23)60+723log⁡(x2+x(2−23)−23+1+223)120−7233atan⁡(2233x3−33+22333)60−RootSum⁡(12t3−12t2−1,(t↦tlog⁡(24t25−18t5+x+35)))10+3RootSum⁡(36t3−6t+1,(t↦tlog⁡(36t2+6t+x−3)))5\int \frac{\left(x^{2} - x\right) + 1}{\left(x^{4} + 2 x^{3}\right) - 3}\, dx = C + \frac{\log{\left(x - 1 \right)}}{10} - \frac{7 \sqrt[3]{2} \log{\left(x + 1 + \sqrt[3]{2} \right)}}{60} + \frac{7 \sqrt[3]{2} \log{\left(x^{2} + x \left(2 - \sqrt[3]{2}\right) - \sqrt[3]{2} + 1 + 2^{\frac{2}{3}} \right)}}{120} - \frac{7 \sqrt[3]{2} \sqrt{3} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3}}{3} \right)}}{60} - \frac{\operatorname{RootSum} {\left(12 t^{3} - 12 t^{2} - 1, \left( t \mapsto t \log{\left(\frac{24 t^{2}}{5} - \frac{18 t}{5} + x + \frac{3}{5} \right)} \right)\right)}}{10} + \frac{3 \operatorname{RootSum} {\left(36 t^{3} - 6 t + 1, \left( t \mapsto t \log{\left(36 t^{2} + 6 t + x - 3 \right)} \right)\right)}}{5}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-10001000
Trazar el gráfico f(x)
Respuesta [src] 
  1                  
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 |                   
 |         2         
 |    1 + x  - x     
 |  -------------- dx
 |        4      3   
 |  -3 + x  + 2*x    
 |                   
/                    
0
∫01x2−x+1x4+2x3−3 dx\int\limits_{0}^{1} \frac{x^{2} - x + 1}{x^{4} + 2 x^{3} - 3}\, dx 
=
= 
  1                  
  /                  
 |                   
 |         2         
 |    1 + x  - x     
 |  -------------- dx
 |        4      3   
 |  -3 + x  + 2*x    
 |                   
/                    
0
∫01x2−x+1x4+2x3−3 dx\int\limits_{0}^{1} \frac{x^{2} - x + 1}{x^{4} + 2 x^{3} - 3}\, dx 
Integral((1 + x^2 - x)/(-3 + x^4 + 2*x^3), (x, 0, 1))
Respuesta numérica [src] 
-4.50418124226594
-4.50418124226594

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

    © Sr Examen – Калькулятор