integrate x/(1+cos(x)^2) from 0 to pi

I thought integration by parts udv = uv - vdu, so x is u and 1/(1+cos(x)^2) is v. But I have trouble finding the trick to integrate v. Thanks for any help or insight

Printable View

- October 24th 2012, 07:28 AMphigirlintegrate x/(1+cos(x)^2) from 0 to pi
integrate x/(1+cos(x)^2) from 0 to pi

I thought integration by parts udv = uv - vdu, so x is u and 1/(1+cos(x)^2) is v. But I have trouble finding the trick to integrate v. Thanks for any help or insight - October 24th 2012, 07:33 AMSironRe: integrate x/(1+cos(x)^2) from 0 to pi
You'll need a numerical integration method to solve this integral.

- October 24th 2012, 09:15 AMJJacquelinRe: integrate x/(1+cos(x)^2) from 0 to pi
Is it (cos(x))^2 or cos(x^2) ?

Is it a school work ? If YES, what exactly is the wording ? - October 24th 2012, 09:18 AMphigirlRe: integrate x/(1+cos(x)^2) from 0 to pi
(cos(x))^2

show the integral x/(1+(cos(x))^2) = (pi^2)/2sqrt(2) - October 24th 2012, 12:53 PMJJacquelinRe: integrate x/(1+cos(x)^2) from 0 to pi
OK. That is clear.

You can solve it thanks to integration by parts with u=x and v(x)=primitive of 1/(1+(cos(x))^2)

A primitive of 1/(1+(cos(x)^2)) is v(x)=(1/sqrt(2))*arctan(tan(x)/sqrt(2))

u'=1. So you should have to integrate u'*v=v, wich is very difficult (involving special functions).

But, you don't need to do it explicitly. Since the function v(x) is periodic, one can see that its definite integal from x=0 to x=pi is equal to 0.

So, only x*v(x) is remaining. But there is a discontinuity at x=pi/2. Since v' is even then x*v is also even, the definite integral is :

2*(x*v(x) at x=pi/2) = 2*(pi/2)*(1/sqrt(2))*arctan(tan(pi/2)/sqrt(2)) = 2*(pi/2)*(1/sqrt(2)*(pi/2) = piČ/(2*sqrt(2))