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

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

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

\int \frac{x \cdot 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

\frac{\partial}{\partial y} (cos (x y^{2}))

x \to - 2 lim (x - 1)

\int sin (x) d sin (x) d x

x \to 3 lim (x^{3} - x^{2} + 15)

y = 0, y = cos (x), [0, \frac{3 π}{2}]

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