∫_0 ^π (x^2 /(1+sinx))dx


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Question Number 162575 by Ar Brandon last updated on 30/Dec/21

$\int_{0}^{π} \frac{x^{2}}{1 + \sin x} d x$
	Answered by phanphuoc last updated on 30/Dec/21

	$y o u c a n p u t x = p i - t$
	Answered by mindispower last updated on 30/Dec/21

	$d u = \frac{1}{1 + s i n (x)} \Rightarrow u = \frac{2 s i n (\frac{x}{2})}{s i n (\frac{x}{2}) + c o s (\frac{x}{2})}$ $I B P \Rightarrow 2 π^{2} - 4 \int_{0}^{π} \frac{s i n (\frac{x}{2})}{c o s (\frac{x}{2}) + \sin (\frac{x}{2})} xdx$ $\int_{0}^{π} \frac{s i n (\frac{x}{2})}{c o s (\frac{x}{2}) + s i n (\frac{x}{2})} x d x = 4 \int_{0}^{\frac{π}{2}} \frac{s i n (y)}{s i n (y) + c o s (y)} y d y$ $= 2 \int_{0}^{\frac{π}{2}} y d y - 2 \int_{0}^{\frac{π}{2}} \frac{c o s (y) - s i n (y)}{s i n (y) + c o s (y)} y d y$ $= \frac{π^{2}}{4} + 2 \int_{0}^{\frac{π}{2}} l n (s i n (y) + c o s (y)) d y$ $\frac{π^{2}}{4} + 2 \int_{0}^{\frac{π}{2}} l n (\sqrt{2}) + l n (s i n (y + \frac{π}{4})) d y$ $= \frac{π^{2}}{4} + π l n (\sqrt{2}) + 2 \int_{0}^{\frac{π}{4}} l n (s i n (y + \frac{π}{4})) d y + 2 \int_{\frac{π}{4}}^{\frac{π}{2}} l n (s i n (y + \frac{π}{4})) d y$ $= \frac{π}{2} (\frac{π}{2} + l n (2)) + 2 \int_{0}^{\frac{π}{4}} l n (c o s (y)) d y + 2 \int_{0}^{\frac{π}{4}} l n (c o s (y)) d y$ $\int_{0}^{\frac{π}{4}} l n (c o s (x)) d x = \frac{1}{4} (2 G - π l n (2))$ $2 G - π \frac{l n (2)}{2} + \frac{π^{2}}{4}$ $\int_{0}^{π} \frac{x^{2} d x}{1 + s i n (x)} = 2 π^{2} - 4 (2 G - \frac{π l n (2)}{2} + \frac{π^{2}}{4})$ $= π^{2} + 2 π l n (2) - 8 G$
	Commented by Ar Brandon last updated on 30/Dec/21

	$Cool! Merci, grand .$
	Commented by mindispower last updated on 31/Dec/21

	$A v e c p l a i s i r B o n n e j o u r n e e$
	Commented by Ar Brandon last updated on 31/Dec/21

	$Meilleure \overset{`}{a} vous!$
	Commented by mindispower last updated on 31/Dec/21

	$t u v a s f a i r e M P m a t h s s p e ?$
	Commented by Ar Brandon last updated on 31/Dec/21

	$Non, je n^{'} ai pas penser \overset{`}{a} le faire .$ $Je suis en 1^{er} ann \overset{´}{e e} IUT actuellement .$ $Et vous ?$
	Commented by mindispower last updated on 31/Dec/21

	$j e s u i s e n M 2 M a t h s f o n d a m e n t a l (T o p o l o g i e A l g e b r i q u e)$ $B o n n e C o n t i n u a t i o n$
	Commented by mindispower last updated on 01/Jan/22

	$b o n j o u r j e v a i s f a i r e u n c o m p t e$ $p o u r e c h a n g e r p a s d e s o u c i s$
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