Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (xe^x) / (x + 1)² dx

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Recognize that the integral is of the form $\int \frac{x e^{x}}{(x+1)^2} \, dx$. Since the denominator is $(x+1)^2$, consider using a substitution to simplify the expression.

Let $u = x + 1$. Then, $x = u - 1$ and $dx = du$. Rewrite the integral in terms of $u$: $\int \frac{(u - 1) e^{u - 1}}{u^2} \, du$.

Rewrite the exponential term as $e^{u - 1} = e^{u} e^{-1} = \frac{e^{u}}{e}$. So the integral becomes $\frac{1}{e} \int \frac{(u - 1) e^{u}}{u^2} \, du$.

Split the integral into two parts: $\frac{1}{e} \int \frac{u e^{u}}{u^2} \, du - \frac{1}{e} \int \frac{e^{u}}{u^2} \, du$, which simplifies to $\frac{1}{e} \int \frac{e^{u}}{u} \, du - \frac{1}{e} \int \frac{e^{u}}{u^2} \, du$.

At this point, consider using integration by parts on the integral $\int \frac{e^{u}}{u^2} \, du$ to simplify it further, choosing appropriate functions for $u$ and $dv$ to reduce the power of $u$ in the denominator.

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