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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

Simplify

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

Add a constant to the solution

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

y^{^{''}} + 4 y^{^{'}} + 3 y = 65 sin (2 t)

f (x) = 3 - x

x^{2} + 1, x, [0, 1]

\frac{\partial}{\partial x} (3 x y^{2} - 2 y + 5 x^{2} y^{2})

f (x) = x + 3, at x = - 2

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