What is the integral of (xe^x)/(2(1+x)^2) ?

The integral of (xe^x)/(2(1+x)^2) is (e^x)/(2(1+x))+C

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integral of (xe^x)/(2(1+x)^2)

\int \frac{x e ^{x}}{2 ( 1 + x ) ^{2}} d x

\frac{e ^{x}}{2 ( 1 + x )} + C

Solution steps

\int \frac{x e ^{x}}{2 ( 1 + x ) ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{x e ^{x}}{( 1 + x ) ^{2}} d x

Apply Integration By Parts

= \frac{1}{2} (- \frac{e ^{x} x}{1 + x} - \int - e^{x} d x)

\int - e^{x} d x = - e^{x}

= \frac{1}{2} (- \frac{e ^{x} x}{1 + x} - (- e^{x}))

Simplify

\frac{1}{2} (- \frac{e ^{x} x}{1 + x} - (- e^{x})) : \frac{e ^{x}}{2 ( 1 + x )}

= \frac{e ^{x}}{2 ( 1 + x )}

Add a constant to the solution

= \frac{e ^{x}}{2 ( 1 + x )} + C

What is the integral of (xe^x)/(2(1+x)^2) ?
The integral of (xe^x)/(2(1+x)^2) is (e^x)/(2(1+x))+C