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Integrate $$\int^{\pi/4}_{0} \frac{x^2}{(x \sin x + \cos x)^2}dx$$

How can I integrate this ?

Question asks us to re-arrange the numerator

Marco Cantarini
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Angelo Mark
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    Hint: write $x^2$ as $x^2(\cos^2x+\sin^2x)$ and then add and subtract $x\cos x\sin x$. Try to look for the derivative of a quotient. – mickep Jul 10 '16 at 14:54

1 Answers1

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If we multiply by $\cos\left(x\right) $ the numerator and the denominator and we integrate by parts we have that $$\begin{align} \int_{0}^{\pi/4}\frac{x^{2}}{\left(x\sin\left(x\right)+\cos\left(x\right)\right)^{2}}dx= & \int_{0}^{\pi/4}\frac{x}{\cos\left(x\right)}\frac{x\cos\left(x\right)}{\left(x\sin\left(x\right)+\cos\left(x\right)\right)^{2}}dx \\ = & -\frac{2\pi}{4+\pi}+\int_{0}^{\pi/4}\frac{1}{\cos^{2}\left(x\right)}dx \\ = &-\frac{2\pi}{4+\pi}+1 \\ = & \color{red}{\frac{4-\pi}{4+\pi}.} \end{align}$$

Marco Cantarini
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    This is a more natural way to calculate this integral than I suggested in the comment. It also shows that the suggested answer in the question is wrong. – mickep Jul 10 '16 at 15:34