Problem

Use the substitution $u=-x$ to evaluate $\int_{-2}^{2} \frac{x^{2}}{e^{x}+1} d x$

Answer

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Answer

\(\boxed{-\int_{2}^{-2} \frac{u^{2}}{e^{u}+1} d u}\)

Steps

Step 1 ：Let $u = -x$. Then, $x = -u$ and $d x = -d u$. Also, when $x = -2$, $u = 2$, and when $x = 2$, $u = -2$.

Step 2 ：Substitute $u$ into the integral: $\int_{-2}^{2} \frac{x^{2}}{e^{x}+1} d x = -\int_{2}^{-2} \frac{u^{2}}{e^{u}+1} d u$.

Step 3 ：Evaluate the integral: $\int_{-2}^{2} \frac{x^{2}}{e^{x}+1} d x = -\int_{2}^{-2} \frac{u^{2}}{e^{u}+1} d u$.

Step 4 ：\(\boxed{-\int_{2}^{-2} \frac{u^{2}}{e^{u}+1} d u}\)

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