Question

here is another question that is bugging me deifinite integral using definite integral

Answer 1

\[\int\limits_{o}^{\pi}x \log(\sin x)= (-\pi^2\div 2)(\log 2)\]

Answer 2

you have to proof that integral is equal to ((-pi^2)/2))*log 2

Answer 3

ew

Answer 4

have you tried using integration by parts?

Answer 5

yeah integration by parts looks like the way to start with my guess is log(sinx) being u

Answer 6

tried but not working using the properties I was able to reduce it to
\[\int\limits_{0}^{\pi}\log \sin x\]
I tried to solve this parts but i am stuck

Answer 7

but is there anyway to make that prettier..it doesnt look so.

Answer 8

@Davidjohn tell me this question has been kicking my butt only one i am not able to :(

Answer 9

let u = log(sin x) and dv = xdx

is this what you did?

Answer 10

your substitution is wrong it would be du=cotxdx

Answer 11

right

Answer 12

it would be cot x...what's the problem with that hehe

Answer 13

ah okay. it just takes a lot of integration by parts but things cancel out nicely. you change halfway through to the u being the x's so they eventually cancel completely