What is the integral of (xe^x)/(sqrt(9+e^x)) ?

The integral of (xe^x)/(sqrt(9+e^x)) is

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integral of (xe^x)/(sqrt(9+e^x))

\int \frac{x e ^{x}}{9 + e ^{x}} d x

2 x 9 + e^{x} + 12 (\frac{1}{2} ln \frac{1}{3} 9 + e^{x} + 1 - \frac{1}{2} ln \frac{1}{3} 9 + e^{x} - 1 - \frac{1}{3} 9 + e^{x}) + C

Solution steps

\int \frac{x e ^{x}}{9 + e ^{x}} d x

Apply u-substitution

: \int \frac{ln ( u )}{9 + u} d u

= \int \frac{ln ( u )}{9 + u} d u

Apply Integration By Parts

: 2 ln (u) 9 + u - \int \frac{2 9 + u}{u} d u

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

\int \frac{2 9 + u}{u} d u = - 12 (\frac{1}{2} ln \frac{9 + u}{3} + 1 - \frac{1}{2} ln \frac{9 + u}{3} - 1 - \frac{1}{3} 9 + u)

= 2 ln (u) 9 + u - (- 12 (\frac{1}{2} ln \frac{9 + u}{3} + 1 - \frac{1}{2} ln \frac{9 + u}{3} - 1 - \frac{1}{3} 9 + u))

Substitute back

u = e^{x}

= 2 ln (e^{x}) 9 + e^{x} - (- 12 (\frac{1}{2} ln \frac{9 + e ^{x}}{3} + 1 - \frac{1}{2} ln \frac{9 + e ^{x}}{3} - 1 - \frac{1}{3} 9 + e^{x}))

Simplify

2 ln (e^{x}) 9 + e^{x} - (- 12 (\frac{1}{2} ln \frac{9 + e ^{x}}{3} + 1 - \frac{1}{2} ln \frac{9 + e ^{x}}{3} - 1 - \frac{1}{3} 9 + e^{x})) : 2 x 9 + e^{x} + 12 (\frac{1}{2} ln \frac{1}{3} 9 + e^{x} + 1 - \frac{1}{2} ln \frac{1}{3} 9 + e^{x} - 1 - \frac{1}{3} 9 + e^{x})

= 2 x 9 + e^{x} + 12 (\frac{1}{2} ln \frac{1}{3} 9 + e^{x} + 1 - \frac{1}{2} ln \frac{1}{3} 9 + e^{x} - 1 - \frac{1}{3} 9 + e^{x})

Add a constant to the solution

= 2 x 9 + e^{x} + 12 (\frac{1}{2} ln \frac{1}{3} 9 + e^{x} + 1 - \frac{1}{2} ln \frac{1}{3} 9 + e^{x} - 1 - \frac{1}{3} 9 + e^{x}) + C

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What is the integral of (xe^x)/(sqrt(9+e^x)) ?
The integral of (xe^x)/(sqrt(9+e^x)) is