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Problem 93 Hard Difficulty

Use Exercise 92 to evaluate the integral
$$ \int^{\pi}_0 \frac{x \sin x}{1 + \cos^2 x} \, dx $$

Answer

$$\frac{\pi^{2}}{4}$$

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Video Transcript

okay. As the question of said, we're using the previous exercise to evaluate this integral. Which means the first thing we can do is we can take Pi over two out and rewrite it. No, we know because that you equivalent to co sign axe, which means negative sign. X d ox is equivalent to gie you. Which means now we have negative power, too. Notice that we added in the negative sign limits of integration have now flip to coast signs here on the bottom, which is one coast on pie on the top, which is negative one we know that do you over one post you squared is the same thing as inverse tangent, actually, so now we actually know we've written are integral, which means we cannot integrate so plug in negative negative becomes positive

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