calculate ∫_(π/4) ^(π/3) ((sinx)/(1+sin^2 x))dx


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Question Number 51989 by maxmathsup by imad last updated on 01/Jan/19

$c a l c u l a t e \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{s i n x}{1 + {s i n}^{2} x} d x$
	Answered by peter frank last updated on 01/Jan/19

	$\int \frac{\sin x}{2 - \cos^{2} x} d x$ $u = \cos x$ $d u = - \sin x d x$ $d x = - \frac{d u}{\sin x}$ $\int \frac{\sin x}{2 - u^{2}} . - \frac{d u}{\sin x}$ $\int - \frac{d u}{2 - u^{2}}$ $- \frac{1}{2 \sqrt{2}} [\int \frac{d u}{\sqrt{2} + u} - \int \frac{d u}{\sqrt{2} - u}]$ $- \frac{1}{2 \sqrt{2}} l n [\frac{\sqrt{2} + u}{\sqrt{2} - u}] + B$ $- \frac{1}{2 \sqrt{2}} l n [\frac{\sqrt{2} + \cos x}{\sqrt{2} - \cos x}] + B$
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