0-pi-x-2-1-sinx-dx-

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