Evaluate the integral: int frac{x e^{2x}}{(1+ | vedclass

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(B) We are given the integral $I = \int \frac{x e^{2x}}{(1+2x)^2} dx$. Let $2x = t$,then $2 dx = dt$,which implies $dx = \frac{dt}{2}$. Substituting t

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

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(B) We are given the integral $I = \int \frac{x e^{2x}}{(1+2x)^2} dx$. Let $2x = t$,then $2 dx = dt$,which implies $dx = \frac{dt}{2}$. Substituting these into the integral: $I = \int \frac{(t/2) e^t}{(1+t)^2} \cdot \frac{dt}{2} = \frac{1}{4} \int \frac{t e^t}{(1+t)^2} dt$. We can rewrite the numerator $t$ as $(t+1-1)$: $I = \frac{1}{4} \int e^t \left( \frac{t+1-1}{(1+t)^2} \right) dt = \frac{1}{4} \int e^t \left( \frac{1}{1+t} - \frac{1}{(1+t)^2} \right) dt$. Using the standard integral formula $\int e^t (f(t) + f'(t)) dt = e^t f(t) + C$,where $f(t) = \frac{1}{1+t}$ and $f'(t) = -\frac{1}{(1+t)^2}$: $I = \frac{1}{4} e^t \left( \frac{1}{1+t} \right) + C$. Substituting $t = 2x$ back: $I = \frac{e^{2x}}{4(1+2x)} + C$.

Evaluate the integral: $\int \frac{x e^{2x}}{(1+2x)^2} dx = $ (where $C$ is a constant of integration.)

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