Integral of x cotan (x) dx from 0 to pi/2

Please solve using all steps. thanks!!

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- May 28th 2010, 10:30 AMcalifsurferboy33Integral of x cotan (x) dx from 0 to pi/2
Integral of x cotan (x) dx from 0 to pi/2

Please solve using all steps. thanks!! - May 28th 2010, 11:31 AMgalactus
Did you try doing this 'under the Integral'?. This one is often done that way.

$\displaystyle xcot(x)=\frac{x}{tan(x)}$

$\displaystyle I(a)=\int_{0}^{\frac{\pi}{2}}\frac{tan^{-1}(a\cdot tan(x))}{tan(x)}dx$

$\displaystyle I'(a)=\frac{d}{dx}\int_{0}^{\frac{\pi}{2}}\frac{ta n^{-1}(a\cdot tan(x))}{tan(x)}dx$

$\displaystyle =\int_{0}^{\frac{\pi}{2}}\frac{\partial}{{\partial }b}\left[\frac{tan^{-1}(a\cdot tan(x))}{tan(x)}\right]dx$

$\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{1}{(a\cdot tan(x))^{2}+1}dx$

$\displaystyle =\frac{\pi}{2(a+1)}$

Now, integrate w.r.t a. Then, let a=1 to finish line.

This is a cool method. Learn about 'differentiating under the integral sign'.

You can probably find it somewhere.