What is the integral of (x*e^{2x})/((2x+1)^2) ?

The integral of (x*e^{2x})/((2x+1)^2) is -(e^{2x}x)/(2(2x+1))+1/4 e^{2x}+C

integral of (x*e^{2x})/((2x+1)^2)

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

- \frac{e ^{2 x} x}{2 ( 2 x + 1 )} + \frac{1}{4} e^{2 x} + C

Solution steps

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

Apply Integration By Parts

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

\int - \frac{1}{2} e^{2 x} d x = - \frac{1}{4} e^{2 x}

= - \frac{e ^{2 x} x}{2 ( 2 x + 1 )} - (- \frac{1}{4} e^{2 x})

Simplify

= - \frac{e ^{2 x} x}{2 ( 2 x + 1 )} + \frac{1}{4} e^{2 x}

Add a constant to the solution

= - \frac{e ^{2 x} x}{2 ( 2 x + 1 )} + \frac{1}{4} e^{2 x} + C

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What is the integral of (x*e^{2x})/((2x+1)^2) ?
The integral of (x*e^{2x})/((2x+1)^2) is -(e^{2x}x)/(2(2x+1))+1/4 e^{2x}+C