Question

Prove that\[\int_{0}^{\pi}\frac{x\sin x}{1+\cos^2x}dx=\frac{\pi^2}{4}\]

Answer 1

Thanks to Zarkon's tips I've got it up until\[-x\tan^{-1}(\cos x)+\int_{0}^{\pi}\tan^{-1}(\cos x)dx\]Which of course I don't know how to integrate.
His tip from here was to use the bounds of integration and the fact that the arctangent is an odd function. Obviously the cosine is even, so arctan(cosx) is even as well I suppose, but he said there is a way to fix that.
Still haven't figured out how to do that...

Answer 2

you should really evaluate the first part from 0 to Pi

Answer 3

\[\frac{\pi^2}{4}+\int_{0}^{\pi}\tan^{-1}(\cos x)dx\]Ok so it looks like we almost have the answer, but for proving that the integral is zero, as mentioned last night.

Answer 4

use the hint i gave yesterday

Answer 5

I woke up thinking about it I promise :)

Answer 6

I mention above the tip you are referring to, right? Arctan odd, fix cosx somehow...

Answer 7

yes...and that the integral of and odd function from -a to a is zero

Answer 8

of course, I'm not that remedial...

Answer 9

:)...sorry

Answer 10

No, I'm sure it seems like it sometimes.
but how to fix cosx...

Answer 11

does it involve using the half tangent formulas?

Answer 12

I've got a much easier method to solve it.

Answer 13

no..simpler :)

Answer 14

convert the integral from 0 to pi TO -pi/2 to pi/2?

Answer 15

yes

Answer 16

Lets substitute \(u=x+\frac{\pi}{2}\), then (call the integral I):
\[I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{(\frac{\pi}{2}-u)\cos(u)}{1+\sin^2u}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\frac{\pi}{2}\cos(u)}{1+\sin^2(u)}.\]

Answer 17

That's because \(\frac{u\cos(u)}{1+\sin^2(u}\) is odd, so its integration from -pi/2 to pi/2 is 0. The rest is trivial.

Answer 18

That's actually more like what I was trying to do, split up the integral, but I couldn't figure out how.

Answer 19

I shifted the limits and use the fact that \[\cos(x+\pi/2)=-\sin(x)\]

which is odd..the composition of odd functions is odd

Answer 20

Typo, the substitution is \(u=\frac{\pi}{2}-x\). The rest is ok I think.

Answer 21

Yes, I could see that was much simpler once asnaseer said it.

Answer 22

Oh asnaseer already said that.

Answer 23

Zarkon is the one who left BIG FOOTPRINT clues all over the place - we just took a long time noticing them :-)

Answer 24

It's so obvious once you know the answer XD

Answer 25

a lot of problem are like that

Answer 26

There's one more I'm gonna post from the calc section of the test. The rest is liner algebra. I don't think I stand a chance with that yet, need to brush up.
Thanks again everybody!