Integral de -(4*x^(3/2)+1)/(x^3+4) dx
Solución
Solución detallada
Hay varias maneras de calcular esta integral.
Método #1
Vuelva a escribir el integrando:
− 4 x 3 2 − 1 x 3 + 4 = − 4 x 3 2 + 1 x 3 + 4 \frac{- 4 x^{\frac{3}{2}} - 1}{x^{3} + 4} = - \frac{4 x^{\frac{3}{2}} + 1}{x^{3} + 4} x 3 + 4 − 4 x 2 3 − 1 = − x 3 + 4 4 x 2 3 + 1
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − 4 x 3 2 + 1 x 3 + 4 ) d x = − ∫ 4 x 3 2 + 1 x 3 + 4 d x \int \left(- \frac{4 x^{\frac{3}{2}} + 1}{x^{3} + 4}\right)\, dx = - \int \frac{4 x^{\frac{3}{2}} + 1}{x^{3} + 4}\, dx ∫ ( − x 3 + 4 4 x 2 3 + 1 ) d x = − ∫ x 3 + 4 4 x 2 3 + 1 d x
Vuelva a escribir el integrando:
4 x 3 2 + 1 x 3 + 4 = 4 x 3 2 x 3 + 4 + 1 x 3 + 4 \frac{4 x^{\frac{3}{2}} + 1}{x^{3} + 4} = \frac{4 x^{\frac{3}{2}}}{x^{3} + 4} + \frac{1}{x^{3} + 4} x 3 + 4 4 x 2 3 + 1 = x 3 + 4 4 x 2 3 + x 3 + 4 1
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 4 x 3 2 x 3 + 4 d x = 4 ∫ x 3 2 x 3 + 4 d x \int \frac{4 x^{\frac{3}{2}}}{x^{3} + 4}\, dx = 4 \int \frac{x^{\frac{3}{2}}}{x^{3} + 4}\, dx ∫ x 3 + 4 4 x 2 3 d x = 4 ∫ x 3 + 4 x 2 3 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
2 2 3 3 log ( − 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 12 − 2 2 3 3 log ( 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 12 + 2 2 3 atan ( 2 2 3 x 2 ) 3 + 2 2 3 atan ( 2 2 3 x − 3 ) 6 + 2 2 3 atan ( 2 2 3 x + 3 ) 6 \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{12} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{6} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{6} 12 2 3 2 3 l o g ( − 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) − 12 2 3 2 3 l o g ( 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) + 3 2 3 2 atan ( 2 2 3 2 x ) + 6 2 3 2 atan ( 2 3 2 x − 3 ) + 6 2 3 2 atan ( 2 3 2 x + 3 )
Por lo tanto, el resultado es: 2 2 3 3 log ( − 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 − 2 2 3 3 log ( 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 + 4 ⋅ 2 2 3 atan ( 2 2 3 x 2 ) 3 + 2 ⋅ 2 2 3 atan ( 2 2 3 x − 3 ) 3 + 2 ⋅ 2 2 3 atan ( 2 2 3 x + 3 ) 3 \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} 3 2 3 2 3 l o g ( − 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) − 3 2 3 2 3 l o g ( 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) + 3 4 ⋅ 2 3 2 atan ( 2 2 3 2 x ) + 3 2 ⋅ 2 3 2 atan ( 2 3 2 x − 3 ) + 3 2 ⋅ 2 3 2 atan ( 2 3 2 x + 3 )
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
2 2 3 log ( x + 2 2 3 ) 12 − 2 2 3 log ( x 2 − 2 2 3 x + 2 2 3 ) 24 + 2 2 3 3 atan ( 2 3 3 x 3 − 3 3 ) 12 \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} 12 2 3 2 l o g ( x + 2 3 2 ) − 24 2 3 2 l o g ( x 2 − 2 3 2 x + 2 3 2 ) + 12 2 3 2 3 atan ( 3 3 2 3 x − 3 3 )
El resultado es: 2 2 3 log ( x + 2 2 3 ) 12 − 2 2 3 log ( x 2 − 2 2 3 x + 2 2 3 ) 24 + 2 2 3 3 log ( − 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 − 2 2 3 3 log ( 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 + 4 ⋅ 2 2 3 atan ( 2 2 3 x 2 ) 3 + 2 ⋅ 2 2 3 atan ( 2 2 3 x − 3 ) 3 + 2 ⋅ 2 2 3 atan ( 2 2 3 x + 3 ) 3 + 2 2 3 3 atan ( 2 3 3 x 3 − 3 3 ) 12 \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} + \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} 12 2 3 2 l o g ( x + 2 3 2 ) − 24 2 3 2 l o g ( x 2 − 2 3 2 x + 2 3 2 ) + 3 2 3 2 3 l o g ( − 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) − 3 2 3 2 3 l o g ( 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) + 3 4 ⋅ 2 3 2 atan ( 2 2 3 2 x ) + 3 2 ⋅ 2 3 2 atan ( 2 3 2 x − 3 ) + 3 2 ⋅ 2 3 2 atan ( 2 3 2 x + 3 ) + 12 2 3 2 3 atan ( 3 3 2 3 x − 3 3 )
Por lo tanto, el resultado es: − 2 2 3 log ( x + 2 2 3 ) 12 + 2 2 3 log ( x 2 − 2 2 3 x + 2 2 3 ) 24 − 2 2 3 3 log ( − 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 + 2 2 3 3 log ( 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 − 4 ⋅ 2 2 3 atan ( 2 2 3 x 2 ) 3 − 2 ⋅ 2 2 3 atan ( 2 2 3 x − 3 ) 3 − 2 ⋅ 2 2 3 atan ( 2 2 3 x + 3 ) 3 − 2 2 3 3 atan ( 2 3 3 x 3 − 3 3 ) 12 - \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} + \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} − 12 2 3 2 l o g ( x + 2 3 2 ) + 24 2 3 2 l o g ( x 2 − 2 3 2 x + 2 3 2 ) − 3 2 3 2 3 l o g ( − 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) + 3 2 3 2 3 l o g ( 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) − 3 4 ⋅ 2 3 2 atan ( 2 2 3 2 x ) − 3 2 ⋅ 2 3 2 atan ( 2 3 2 x − 3 ) − 3 2 ⋅ 2 3 2 atan ( 2 3 2 x + 3 ) − 12 2 3 2 3 atan ( 3 3 2 3 x − 3 3 )
Método #2
Vuelva a escribir el integrando:
− 4 x 3 2 − 1 x 3 + 4 = − 4 x 3 2 x 3 + 4 − 1 x 3 + 4 \frac{- 4 x^{\frac{3}{2}} - 1}{x^{3} + 4} = - \frac{4 x^{\frac{3}{2}}}{x^{3} + 4} - \frac{1}{x^{3} + 4} x 3 + 4 − 4 x 2 3 − 1 = − x 3 + 4 4 x 2 3 − x 3 + 4 1
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − 4 x 3 2 x 3 + 4 ) d x = − 4 ∫ x 3 2 x 3 + 4 d x \int \left(- \frac{4 x^{\frac{3}{2}}}{x^{3} + 4}\right)\, dx = - 4 \int \frac{x^{\frac{3}{2}}}{x^{3} + 4}\, dx ∫ ( − x 3 + 4 4 x 2 3 ) d x = − 4 ∫ x 3 + 4 x 2 3 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
2 2 3 3 log ( − 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 12 − 2 2 3 3 log ( 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 12 + 2 2 3 atan ( 2 2 3 x 2 ) 3 + 2 2 3 atan ( 2 2 3 x − 3 ) 6 + 2 2 3 atan ( 2 2 3 x + 3 ) 6 \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{12} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{6} + \frac{2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{6} 12 2 3 2 3 l o g ( − 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) − 12 2 3 2 3 l o g ( 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) + 3 2 3 2 atan ( 2 2 3 2 x ) + 6 2 3 2 atan ( 2 3 2 x − 3 ) + 6 2 3 2 atan ( 2 3 2 x + 3 )
Por lo tanto, el resultado es: − 2 2 3 3 log ( − 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 + 2 2 3 3 log ( 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 − 4 ⋅ 2 2 3 atan ( 2 2 3 x 2 ) 3 − 2 ⋅ 2 2 3 atan ( 2 2 3 x − 3 ) 3 − 2 ⋅ 2 2 3 atan ( 2 2 3 x + 3 ) 3 - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} − 3 2 3 2 3 l o g ( − 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) + 3 2 3 2 3 l o g ( 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) − 3 4 ⋅ 2 3 2 atan ( 2 2 3 2 x ) − 3 2 ⋅ 2 3 2 atan ( 2 3 2 x − 3 ) − 3 2 ⋅ 2 3 2 atan ( 2 3 2 x + 3 )
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − 1 x 3 + 4 ) d x = − ∫ 1 x 3 + 4 d x \int \left(- \frac{1}{x^{3} + 4}\right)\, dx = - \int \frac{1}{x^{3} + 4}\, dx ∫ ( − x 3 + 4 1 ) d x = − ∫ x 3 + 4 1 d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
2 2 3 log ( x + 2 2 3 ) 12 − 2 2 3 log ( x 2 − 2 2 3 x + 2 2 3 ) 24 + 2 2 3 3 atan ( 2 3 3 x 3 − 3 3 ) 12 \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} - \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} 12 2 3 2 l o g ( x + 2 3 2 ) − 24 2 3 2 l o g ( x 2 − 2 3 2 x + 2 3 2 ) + 12 2 3 2 3 atan ( 3 3 2 3 x − 3 3 )
Por lo tanto, el resultado es: − 2 2 3 log ( x + 2 2 3 ) 12 + 2 2 3 log ( x 2 − 2 2 3 x + 2 2 3 ) 24 − 2 2 3 3 atan ( 2 3 3 x 3 − 3 3 ) 12 - \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} + \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} − 12 2 3 2 l o g ( x + 2 3 2 ) + 24 2 3 2 l o g ( x 2 − 2 3 2 x + 2 3 2 ) − 12 2 3 2 3 atan ( 3 3 2 3 x − 3 3 )
El resultado es: − 2 2 3 log ( x + 2 2 3 ) 12 + 2 2 3 log ( x 2 − 2 2 3 x + 2 2 3 ) 24 − 2 2 3 3 log ( − 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 + 2 2 3 3 log ( 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 − 4 ⋅ 2 2 3 atan ( 2 2 3 x 2 ) 3 − 2 ⋅ 2 2 3 atan ( 2 2 3 x − 3 ) 3 − 2 ⋅ 2 2 3 atan ( 2 2 3 x + 3 ) 3 − 2 2 3 3 atan ( 2 3 3 x 3 − 3 3 ) 12 - \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} + \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} − 12 2 3 2 l o g ( x + 2 3 2 ) + 24 2 3 2 l o g ( x 2 − 2 3 2 x + 2 3 2 ) − 3 2 3 2 3 l o g ( − 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) + 3 2 3 2 3 l o g ( 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) − 3 4 ⋅ 2 3 2 atan ( 2 2 3 2 x ) − 3 2 ⋅ 2 3 2 atan ( 2 3 2 x − 3 ) − 3 2 ⋅ 2 3 2 atan ( 2 3 2 x + 3 ) − 12 2 3 2 3 atan ( 3 3 2 3 x − 3 3 )
Ahora simplificar:
2 2 3 ( − 2 log ( x + 2 2 3 ) + log ( x 2 − 2 2 3 x + 2 2 3 ) − 8 3 log ( − 2 3 3 x + x + 2 2 3 ) + 8 3 log ( 2 3 3 x + x + 2 2 3 ) − 32 atan ( 2 2 3 x 2 ) − 2 3 atan ( 3 ( 2 3 x − 1 ) 3 ) − 16 atan ( 2 2 3 x − 3 ) − 16 atan ( 2 2 3 x + 3 ) ) 24 \frac{2^{\frac{2}{3}} \left(- 2 \log{\left(x + 2^{\frac{2}{3}} \right)} + \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} - 8 \sqrt{3} \log{\left(- \sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} + 8 \sqrt{3} \log{\left(\sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} - 32 \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)} - 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}\right)}{24} 24 2 3 2 ( − 2 l o g ( x + 2 3 2 ) + l o g ( x 2 − 2 3 2 x + 2 3 2 ) − 8 3 l o g ( − 3 2 3 x + x + 2 3 2 ) + 8 3 l o g ( 3 2 3 x + x + 2 3 2 ) − 32 atan ( 2 2 3 2 x ) − 2 3 atan ( 3 3 ( 3 2 x − 1 ) ) − 16 atan ( 2 3 2 x − 3 ) − 16 atan ( 2 3 2 x + 3 ) )
Añadimos la constante de integración:
2 2 3 ( − 2 log ( x + 2 2 3 ) + log ( x 2 − 2 2 3 x + 2 2 3 ) − 8 3 log ( − 2 3 3 x + x + 2 2 3 ) + 8 3 log ( 2 3 3 x + x + 2 2 3 ) − 32 atan ( 2 2 3 x 2 ) − 2 3 atan ( 3 ( 2 3 x − 1 ) 3 ) − 16 atan ( 2 2 3 x − 3 ) − 16 atan ( 2 2 3 x + 3 ) ) 24 + c o n s t a n t \frac{2^{\frac{2}{3}} \left(- 2 \log{\left(x + 2^{\frac{2}{3}} \right)} + \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} - 8 \sqrt{3} \log{\left(- \sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} + 8 \sqrt{3} \log{\left(\sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} - 32 \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)} - 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}\right)}{24}+ \mathrm{constant} 24 2 3 2 ( − 2 l o g ( x + 2 3 2 ) + l o g ( x 2 − 2 3 2 x + 2 3 2 ) − 8 3 l o g ( − 3 2 3 x + x + 2 3 2 ) + 8 3 l o g ( 3 2 3 x + x + 2 3 2 ) − 32 atan ( 2 2 3 2 x ) − 2 3 atan ( 3 3 ( 3 2 x − 1 ) ) − 16 atan ( 2 3 2 x − 3 ) − 16 atan ( 2 3 2 x + 3 ) ) + constant
Respuesta:
2 2 3 ( − 2 log ( x + 2 2 3 ) + log ( x 2 − 2 2 3 x + 2 2 3 ) − 8 3 log ( − 2 3 3 x + x + 2 2 3 ) + 8 3 log ( 2 3 3 x + x + 2 2 3 ) − 32 atan ( 2 2 3 x 2 ) − 2 3 atan ( 3 ( 2 3 x − 1 ) 3 ) − 16 atan ( 2 2 3 x − 3 ) − 16 atan ( 2 2 3 x + 3 ) ) 24 + c o n s t a n t \frac{2^{\frac{2}{3}} \left(- 2 \log{\left(x + 2^{\frac{2}{3}} \right)} + \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)} - 8 \sqrt{3} \log{\left(- \sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} + 8 \sqrt{3} \log{\left(\sqrt[3]{2} \sqrt{3} \sqrt{x} + x + 2^{\frac{2}{3}} \right)} - 32 \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)} - 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(\sqrt[3]{2} x - 1\right)}{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)} - 16 \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}\right)}{24}+ \mathrm{constant} 24 2 3 2 ( − 2 l o g ( x + 2 3 2 ) + l o g ( x 2 − 2 3 2 x + 2 3 2 ) − 8 3 l o g ( − 3 2 3 x + x + 2 3 2 ) + 8 3 l o g ( 3 2 3 x + x + 2 3 2 ) − 32 atan ( 2 2 3 2 x ) − 2 3 atan ( 3 3 ( 3 2 x − 1 ) ) − 16 atan ( 2 3 2 x − 3 ) − 16 atan ( 2 3 2 x + 3 ) ) + constant
Respuesta (Indefinida)
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| - 4*x - 1 \ 2 / 2*2 *atan\\/ 3 + 2 *\/ x / 2*2 *atan\- \/ 3 + 2 *\/ x / 2 *log\x + 2 / 2 *log\x + 2*\/ 2 - x*2 / 2 *\/ 3 *log\4*x + 4*2 - 4*\/ 2 *\/ 3 *\/ x / \ 3 3 / 2 *\/ 3 *log\4*x + 4*2 + 4*\/ 2 *\/ 3 *\/ x /
| ------------ dx = C - ----------------------- - ------------------------------- - --------------------------------- - ------------------ + ------------------------------- - -------------------------------------------------- - ---------------------------------------- + --------------------------------------------------
| 3 3 3 3 12 24 3 12 3
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∫ − 4 x 3 2 − 1 x 3 + 4 d x = C − 2 2 3 log ( x + 2 2 3 ) 12 + 2 2 3 log ( x 2 − 2 2 3 x + 2 2 3 ) 24 − 2 2 3 3 log ( − 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 + 2 2 3 3 log ( 4 2 3 3 x + 4 x + 4 ⋅ 2 2 3 ) 3 − 4 ⋅ 2 2 3 atan ( 2 2 3 x 2 ) 3 − 2 ⋅ 2 2 3 atan ( 2 2 3 x − 3 ) 3 − 2 ⋅ 2 2 3 atan ( 2 2 3 x + 3 ) 3 − 2 2 3 3 atan ( 2 3 3 x 3 − 3 3 ) 12 \int \frac{- 4 x^{\frac{3}{2}} - 1}{x^{3} + 4}\, dx = C - \frac{2^{\frac{2}{3}} \log{\left(x + 2^{\frac{2}{3}} \right)}}{12} + \frac{2^{\frac{2}{3}} \log{\left(x^{2} - 2^{\frac{2}{3}} x + 2 \sqrt[3]{2} \right)}}{24} - \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(- 4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \log{\left(4 \sqrt[3]{2} \sqrt{3} \sqrt{x} + 4 x + 4 \cdot 2^{\frac{2}{3}} \right)}}{3} - \frac{4 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(\frac{2^{\frac{2}{3}} \sqrt{x}}{2} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} - \sqrt{3} \right)}}{3} - \frac{2 \cdot 2^{\frac{2}{3}} \operatorname{atan}{\left(2^{\frac{2}{3}} \sqrt{x} + \sqrt{3} \right)}}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt[3]{2} \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right)}}{12} ∫ x 3 + 4 − 4 x 2 3 − 1 d x = C − 12 2 3 2 log ( x + 2 3 2 ) + 24 2 3 2 log ( x 2 − 2 3 2 x + 2 3 2 ) − 3 2 3 2 3 log ( − 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) + 3 2 3 2 3 log ( 4 3 2 3 x + 4 x + 4 ⋅ 2 3 2 ) − 3 4 ⋅ 2 3 2 atan ( 2 2 3 2 x ) − 3 2 ⋅ 2 3 2 atan ( 2 3 2 x − 3 ) − 3 2 ⋅ 2 3 2 atan ( 2 3 2 x + 3 ) − 12 2 3 2 3 atan ( 3 3 2 3 x − 3 3 )
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.