integral of (xe^x)/(sqrt(81+e^x))

解答

\int \frac{x e ^{x}}{81 + e ^{x}} d x

2 x 81 + e^{x} + 36 (\frac{1}{2} ln \frac{1}{9} 81 + e^{x} + 1 - \frac{1}{2} ln \frac{1}{9} 81 + e^{x} - 1 - \frac{1}{9} 81 + e^{x}) + C

求解步骤

\int \frac{x e ^{x}}{81 + e ^{x}} d x

使用换元积分法

= \int \frac{ln ( u )}{81 + u} d u

使用分布积分法

= 2 ln (u) 81 + u - \int \frac{2 81 + u}{u} d u

\int \frac{2 81 + u}{u} d u = - 36 (\frac{1}{2} ln \frac{81 + u}{9} + 1 - \frac{1}{2} ln \frac{81 + u}{9} - 1 - \frac{1}{9} 81 + u)

= 2 ln (u) 81 + u - (- 36 (\frac{1}{2} ln \frac{81 + u}{9} + 1 - \frac{1}{2} ln \frac{81 + u}{9} - 1 - \frac{1}{9} 81 + u))

u = e^{x}

代回

= 2 ln (e^{x}) 81 + e^{x} - (- 36 (\frac{1}{2} ln \frac{81 + e ^{x}}{9} + 1 - \frac{1}{2} ln \frac{81 + e ^{x}}{9} - 1 - \frac{1}{9} 81 + e^{x}))

化简

2 ln (e^{x}) 81 + e^{x} - (- 36 (\frac{1}{2} ln \frac{81 + e ^{x}}{9} + 1 - \frac{1}{2} ln \frac{81 + e ^{x}}{9} - 1 - \frac{1}{9} 81 + e^{x})) : 2 x 81 + e^{x} + 36 (\frac{1}{2} ln \frac{1}{9} 81 + e^{x} + 1 - \frac{1}{2} ln \frac{1}{9} 81 + e^{x} - 1 - \frac{1}{9} 81 + e^{x})

= 2 x 81 + e^{x} + 36 (\frac{1}{2} ln \frac{1}{9} 81 + e^{x} + 1 - \frac{1}{2} ln \frac{1}{9} 81 + e^{x} - 1 - \frac{1}{9} 81 + e^{x})

解答补常数

= 2 x 81 + e^{x} + 36 (\frac{1}{2} ln \frac{1}{9} 81 + e^{x} + 1 - \frac{1}{2} ln \frac{1}{9} 81 + e^{x} - 1 - \frac{1}{9} 81 + e^{x}) + C

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