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int_(0)^(pi)(x sin x)/(1+cos^(2)x)dx

$\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x$

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Evaluate definite integrals using the chain rule

Full solution

Q.

\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x

Substitution Simplification: Let's use substitution to simplify the integral. Let $u = \cos(x)$ , then $du = -\sin(x)dx$ .
Change Limits of Integration: Change the limits of integration. When $x = 0$ , $u = \cos(0) = 1$ . When $x = \pi$ , $u = \cos(\pi) = -1$ .
Rewrite Integral in terms of u: Rewrite the integral in terms of u. The integral becomes $-\int_{1}^{-1}\frac{x}{1+u^2}\,du$ . But we need to express $x$ in terms of $u$ , which we haven't done yet. This is a mistake, we can't integrate with respect to $u$ without changing $x$ to a function of $u$ .

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