∫_0 ^( (π/2)) x.sin^( 2) (x).ln(sin(x))dx


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Question Number 155252 by mnjuly1970 last updated on 27/Sep/21

$\int_{0}^{\frac{π}{2}} x . {s i n}^{2} (x) . l n (s i n (x)) d x$
	Answered by phanphuoc last updated on 28/Sep/21

	$l n s i n x = - l n 2 - \sum_{k = 1}^{\infty} \frac{c o s (2 k x)}{k}$ $\to I = \int_{0}^{\infty} {x s i n}^{2} x (- l n 2 - Σ \frac{c o s (2 k x)}{k}) d x$ $= - l n 2 \int_{0}^{\infty} {x s i n}^{2} x d x - Σ \frac{1}{2 k} x (1 - c o s 2 x) c o s (2 k x)$ $y o u c a n I B P \to . . . . . .$
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