Integrate:-
$$ \int_0^{\pi/2} \frac{x}{\sin x}\mathrm{d}x$$

Question

Integrate:-
$$ \int_0^{\pi/2} \frac{x}{\sin x}\mathrm{d}x$$

geerky42 · Answer

Quest guess is to change 1/sinx to cscx, then use integration by parts.

dumbcow · Answer

couldn't you use integration by parts here...

u = x            dv = csc x
du = dx         v = ln(csc - cot)

then integrating the log...hmm

experimentx · Answer

did you check if Mathematica returns elementary result for indefinite integral?

mathslover · Answer

Hint : Find the integral

dumbcow · Answer

no i always see if i can do it by hand first, then check wolfram ....

yeah cant be done for indefinite integral, i suggest using simpsons rule to approximate the definite integral

experimentx · Answer

the value of integral seems pretty nice ...  2 times catlan constant.

dumbcow · Answer

interesting, weird coincidence? Or does someone know the significance of that

experimentx · Answer

sometimes this constant comes in improper integral, that I know that ... since I don't have much experience with combinatorics i don't know the significance of it.

anonymous · Answer

it's nice to see you linkha :-)

experimentx · Answer

yep ... you are active after very long time :)

perl · Answer

what did wolfram say the integral is

anonymous · Answer

how do i know

anonymous · Answer

Section 2 gives a list of identities - the fourth is your integral. Section 10 gives a nice proof. http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=FA3716933C799BCB853C3D4F7AD727A1?doi=10.1.1.26.1879&rep=rep1&type=pdf

experimentx · Answer

very interesting link!!

perl · Answer

*eyes glaze over pdf*

anonymous · Answer

I had a feeling a series would be involved, but no idea how/where to apply it...

experimentx · Answer

I was wondering if ... you could expand csc(x) in terms of Bernoulli's number and integrate term by term, it didn't sound nice so gave up the idea.

experimentx · Answer

the proof no 4 is very nice. follows in couple of steps.