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Resolución de integrales/ 3x²+1/ (3x²+1dx)/(3√(2x³+2x+2)²)

Integral de (3x²+1dx)/(3√(2x³+2x+2)²) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                          
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 |  3*\/  2*x  + 2*x + 2     
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0
∫013x2+13((2x3+2x)+2)2 dx\int\limits_{0}^{1} \frac{3 x^{2} + 1}{3 \left(\sqrt{\left(2 x^{3} + 2 x\right) + 2}\right)^{2}}\, dx 
Integral((3*x^2 + 1)/((3*(sqrt(2*x^3 + 2*x + 2))^2)), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    3x2+13((2x3+2x)+2)2=x2(2x3+2x)+2+13((2x3+2x)+2)\frac{3 x^{2} + 1}{3 \left(\sqrt{\left(2 x^{3} + 2 x\right) + 2}\right)^{2}} = \frac{x^{2}}{\left(2 x^{3} + 2 x\right) + 2} + \frac{1}{3 \left(\left(2 x^{3} + 2 x\right) + 2\right)}

  2. Integramos término a término:

    1. Vuelva a escribir el integrando:

      x2(2x3+2x)+2=x22(x3+x+1)\frac{x^{2}}{\left(2 x^{3} + 2 x\right) + 2} = \frac{x^{2}}{2 \left(x^{3} + x + 1\right)}

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

      ∫x22(x3+x+1) dx=∫x2x3+x+1 dx2\int \frac{x^{2}}{2 \left(x^{3} + x + 1\right)}\, dx = \frac{\int \frac{x^{2}}{x^{3} + x + 1}\, dx}{2}

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

        Pero la integral

        RootSum⁡(31t3−31t2+10t−1,(t↦tlog⁡(−62t2+31t+x−3)))\operatorname{RootSum} {\left(31 t^{3} - 31 t^{2} + 10 t - 1, \left( t \mapsto t \log{\left(- 62 t^{2} + 31 t + x - 3 \right)} \right)\right)}

      Por lo tanto, el resultado es: RootSum⁡(31t3−31t2+10t−1,(t↦tlog⁡(−62t2+31t+x−3)))2\frac{\operatorname{RootSum} {\left(31 t^{3} - 31 t^{2} + 10 t - 1, \left( t \mapsto t \log{\left(- 62 t^{2} + 31 t + x - 3 \right)} \right)\right)}}{2}

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

      ∫13((2x3+2x)+2) dx=∫1(2x3+2x)+2 dx3\int \frac{1}{3 \left(\left(2 x^{3} + 2 x\right) + 2\right)}\, dx = \frac{\int \frac{1}{\left(2 x^{3} + 2 x\right) + 2}\, dx}{3}

      1. Vuelva a escribir el integrando:

        1(2x3+2x)+2=12(x3+x+1)\frac{1}{\left(2 x^{3} + 2 x\right) + 2} = \frac{1}{2 \left(x^{3} + x + 1\right)}

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

        ∫12(x3+x+1) dx=∫1x3+x+1 dx2\int \frac{1}{2 \left(x^{3} + x + 1\right)}\, dx = \frac{\int \frac{1}{x^{3} + x + 1}\, dx}{2}

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

          Pero la integral

          RootSum⁡(31t3−3t−1,(t↦tlog⁡(−62t29+31t9+x+49)))\operatorname{RootSum} {\left(31 t^{3} - 3 t - 1, \left( t \mapsto t \log{\left(- \frac{62 t^{2}}{9} + \frac{31 t}{9} + x + \frac{4}{9} \right)} \right)\right)}

        Por lo tanto, el resultado es: RootSum⁡(31t3−3t−1,(t↦tlog⁡(−62t29+31t9+x+49)))2\frac{\operatorname{RootSum} {\left(31 t^{3} - 3 t - 1, \left( t \mapsto t \log{\left(- \frac{62 t^{2}}{9} + \frac{31 t}{9} + x + \frac{4}{9} \right)} \right)\right)}}{2}

      Por lo tanto, el resultado es: RootSum⁡(31t3−3t−1,(t↦tlog⁡(−62t29+31t9+x+49)))6\frac{\operatorname{RootSum} {\left(31 t^{3} - 3 t - 1, \left( t \mapsto t \log{\left(- \frac{62 t^{2}}{9} + \frac{31 t}{9} + x + \frac{4}{9} \right)} \right)\right)}}{6}

    El resultado es: RootSum⁡(31t3−3t−1,(t↦tlog⁡(−62t29+31t9+x+49)))6+RootSum⁡(31t3−31t2+10t−1,(t↦tlog⁡(−62t2+31t+x−3)))2\frac{\operatorname{RootSum} {\left(31 t^{3} - 3 t - 1, \left( t \mapsto t \log{\left(- \frac{62 t^{2}}{9} + \frac{31 t}{9} + x + \frac{4}{9} \right)} \right)\right)}}{6} + \frac{\operatorname{RootSum} {\left(31 t^{3} - 31 t^{2} + 10 t - 1, \left( t \mapsto t \log{\left(- 62 t^{2} + 31 t + x - 3 \right)} \right)\right)}}{2}

  3. Ahora simplificar:

    ((−12−3i2)3931922+1623+131(−12−3i2)3931922+1623)log⁡(x+49+31(−12−3i2)3931922+16239−62((−12−3i2)3931922+1623+131(−12−3i2)3931922+1623)29+19(−12−3i2)3931922+1623)6+(131(−12+3i2)3931922+1623+(−12+3i2)3931922+1623)log⁡(x+49+19(−12+3i2)3931922+1623−62(131(−12+3i2)3931922+1623+(−12+3i2)3931922+1623)29+31(−12+3i2)3931922+16239)6+(13−(−12+3i2)3931922+16233−193(−12+3i2)3931922+1623)log⁡(x+223−31(−12+3i2)3931922+16233−13(−12+3i2)3931922+1623−62(13−(−12+3i2)3931922+16233−193(−12+3i2)3931922+1623)2)2+(13−193(−12−3i2)3931922+1623−(−12−3i2)3931922+16233)log⁡(x+223−62(13−193(−12−3i2)3931922+1623−(−12−3i2)3931922+16233)2−13(−12−3i2)3931922+1623−31(−12−3i2)3931922+16233)2+(−3931922+16233−1933931922+1623+13)log⁡(x−313931922+16233−62(−3931922+16233−1933931922+1623+13)2−133931922+1623+223)2+(1313931922+1623+3931922+1623)log⁡(x−62(1313931922+1623+3931922+1623)29+193931922+1623+49+313931922+16239)6\frac{\left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}} + \frac{1}{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right) \log{\left(x + \frac{4}{9} + \frac{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} - \frac{62 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}} + \frac{1}{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right)^{2}}{9} + \frac{1}{9 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} \right)}}{6} + \frac{\left(\frac{1}{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right) \log{\left(x + \frac{4}{9} + \frac{1}{9 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{62 \left(\frac{1}{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right)^{2}}{9} + \frac{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} \right)}}{6} + \frac{\left(\frac{1}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right) \log{\left(x + \frac{22}{3} - \frac{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - 62 \left(\frac{1}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right)^{2} \right)}}{2} + \frac{\left(\frac{1}{3} - \frac{1}{93 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3}\right) \log{\left(x + \frac{22}{3} - 62 \left(\frac{1}{3} - \frac{1}{93 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3}\right)^{2} - \frac{1}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} \right)}}{2} + \frac{\left(- \frac{\sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{1}{3}\right) \log{\left(x - \frac{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - 62 \left(- \frac{\sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{1}{3}\right)^{2} - \frac{1}{3 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{22}{3} \right)}}{2} + \frac{\left(\frac{1}{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right) \log{\left(x - \frac{62 \left(\frac{1}{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right)^{2}}{9} + \frac{1}{9 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{4}{9} + \frac{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} \right)}}{6}

  4. Añadimos la constante de integración:

    ((−12−3i2)3931922+1623+131(−12−3i2)3931922+1623)log⁡(x+49+31(−12−3i2)3931922+16239−62((−12−3i2)3931922+1623+131(−12−3i2)3931922+1623)29+19(−12−3i2)3931922+1623)6+(131(−12+3i2)3931922+1623+(−12+3i2)3931922+1623)log⁡(x+49+19(−12+3i2)3931922+1623−62(131(−12+3i2)3931922+1623+(−12+3i2)3931922+1623)29+31(−12+3i2)3931922+16239)6+(13−(−12+3i2)3931922+16233−193(−12+3i2)3931922+1623)log⁡(x+223−31(−12+3i2)3931922+16233−13(−12+3i2)3931922+1623−62(13−(−12+3i2)3931922+16233−193(−12+3i2)3931922+1623)2)2+(13−193(−12−3i2)3931922+1623−(−12−3i2)3931922+16233)log⁡(x+223−62(13−193(−12−3i2)3931922+1623−(−12−3i2)3931922+16233)2−13(−12−3i2)3931922+1623−31(−12−3i2)3931922+16233)2+(−3931922+16233−1933931922+1623+13)log⁡(x−313931922+16233−62(−3931922+16233−1933931922+1623+13)2−133931922+1623+223)2+(1313931922+1623+3931922+1623)log⁡(x−62(1313931922+1623+3931922+1623)29+193931922+1623+49+313931922+16239)6+constant\frac{\left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}} + \frac{1}{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right) \log{\left(x + \frac{4}{9} + \frac{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} - \frac{62 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}} + \frac{1}{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right)^{2}}{9} + \frac{1}{9 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} \right)}}{6} + \frac{\left(\frac{1}{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right) \log{\left(x + \frac{4}{9} + \frac{1}{9 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{62 \left(\frac{1}{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right)^{2}}{9} + \frac{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} \right)}}{6} + \frac{\left(\frac{1}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right) \log{\left(x + \frac{22}{3} - \frac{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - 62 \left(\frac{1}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right)^{2} \right)}}{2} + \frac{\left(\frac{1}{3} - \frac{1}{93 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3}\right) \log{\left(x + \frac{22}{3} - 62 \left(\frac{1}{3} - \frac{1}{93 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3}\right)^{2} - \frac{1}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} \right)}}{2} + \frac{\left(- \frac{\sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{1}{3}\right) \log{\left(x - \frac{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - 62 \left(- \frac{\sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{1}{3}\right)^{2} - \frac{1}{3 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{22}{3} \right)}}{2} + \frac{\left(\frac{1}{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right) \log{\left(x - \frac{62 \left(\frac{1}{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right)^{2}}{9} + \frac{1}{9 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{4}{9} + \frac{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} \right)}}{6}+ \mathrm{constant}

Respuesta:

((−12−3i2)3931922+1623+131(−12−3i2)3931922+1623)log⁡(x+49+31(−12−3i2)3931922+16239−62((−12−3i2)3931922+1623+131(−12−3i2)3931922+1623)29+19(−12−3i2)3931922+1623)6+(131(−12+3i2)3931922+1623+(−12+3i2)3931922+1623)log⁡(x+49+19(−12+3i2)3931922+1623−62(131(−12+3i2)3931922+1623+(−12+3i2)3931922+1623)29+31(−12+3i2)3931922+16239)6+(13−(−12+3i2)3931922+16233−193(−12+3i2)3931922+1623)log⁡(x+223−31(−12+3i2)3931922+16233−13(−12+3i2)3931922+1623−62(13−(−12+3i2)3931922+16233−193(−12+3i2)3931922+1623)2)2+(13−193(−12−3i2)3931922+1623−(−12−3i2)3931922+16233)log⁡(x+223−62(13−193(−12−3i2)3931922+1623−(−12−3i2)3931922+16233)2−13(−12−3i2)3931922+1623−31(−12−3i2)3931922+16233)2+(−3931922+16233−1933931922+1623+13)log⁡(x−313931922+16233−62(−3931922+16233−1933931922+1623+13)2−133931922+1623+223)2+(1313931922+1623+3931922+1623)log⁡(x−62(1313931922+1623+3931922+1623)29+193931922+1623+49+313931922+16239)6+constant\frac{\left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}} + \frac{1}{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right) \log{\left(x + \frac{4}{9} + \frac{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} - \frac{62 \left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}} + \frac{1}{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right)^{2}}{9} + \frac{1}{9 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} \right)}}{6} + \frac{\left(\frac{1}{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right) \log{\left(x + \frac{4}{9} + \frac{1}{9 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{62 \left(\frac{1}{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right)^{2}}{9} + \frac{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} \right)}}{6} + \frac{\left(\frac{1}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right) \log{\left(x + \frac{22}{3} - \frac{31 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - 62 \left(\frac{1}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}\right)^{2} \right)}}{2} + \frac{\left(\frac{1}{3} - \frac{1}{93 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3}\right) \log{\left(x + \frac{22}{3} - 62 \left(\frac{1}{3} - \frac{1}{93 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3}\right)^{2} - \frac{1}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} - \frac{31 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} \right)}}{2} + \frac{\left(- \frac{\sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{1}{3}\right) \log{\left(x - \frac{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - 62 \left(- \frac{\sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{3} - \frac{1}{93 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{1}{3}\right)^{2} - \frac{1}{3 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{22}{3} \right)}}{2} + \frac{\left(\frac{1}{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right) \log{\left(x - \frac{62 \left(\frac{1}{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}\right)^{2}}{9} + \frac{1}{9 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}} + \frac{4}{9} + \frac{31 \sqrt[3]{\frac{3 \sqrt{93}}{1922} + \frac{1}{62}}}{9} \right)}}{6}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                              /                           /            2       \\
 |                                                                                                               |    3                      |4       62*t    31*t||
 |           2                            /    3       2                       /             2       \\   RootSum|31*t  - 3*t - 1, t -> t*log|- + x - ----- + ----||
 |        3*x  + 1                 RootSum\31*t  - 31*t  + 10*t - 1, t -> t*log\-3 + x - 62*t  + 31*t//          \                           \9         9      9  //
 | ---------------------- dx = C + -------------------------------------------------------------------- + ----------------------------------------------------------
 |                      2                                           2                                                                 6                             
 |      ________________                                                                                                                                            
 |     /    3                                                                                                                                                       
 | 3*\/  2*x  + 2*x + 2                                                                                                                                             
 |                                                                                                                                                                  
/
∫3x2+13((2x3+2x)+2)2 dx=C+RootSum⁡(31t3−3t−1,(t↦tlog⁡(−62t29+31t9+x+49)))6+RootSum⁡(31t3−31t2+10t−1,(t↦tlog⁡(−62t2+31t+x−3)))2\int \frac{3 x^{2} + 1}{3 \left(\sqrt{\left(2 x^{3} + 2 x\right) + 2}\right)^{2}}\, dx = C + \frac{\operatorname{RootSum} {\left(31 t^{3} - 3 t - 1, \left( t \mapsto t \log{\left(- \frac{62 t^{2}}{9} + \frac{31 t}{9} + x + \frac{4}{9} \right)} \right)\right)}}{6} + \frac{\operatorname{RootSum} {\left(31 t^{3} - 31 t^{2} + 10 t - 1, \left( t \mapsto t \log{\left(- 62 t^{2} + 31 t + x - 3 \right)} \right)\right)}}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.000.25
Trazar el gráfico f(x)
Respuesta [src] 
log(3)
------
  6
log⁡(3)6\frac{\log{\left(3 \right)}}{6} 
=
= 
log(3)
------
  6
log⁡(3)6\frac{\log{\left(3 \right)}}{6} 
log(3)/6
Respuesta numérica [src] 
0.183102048111352
0.183102048111352

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

    © Sr Examen – Калькулятор