Another interesting integral

**simplependulum** · June 6th 2009, 07:50 PM

evaluate the integral $\int_{0}^{\frac{\pi}{2}} x \cot(x) ~dx$

In fact , it is not a challenge problem for an experienced integrator
but i believe that this may help remind us of some techniques

**NonCommAlg** · June 6th 2009, 08:41 PM

Originally Posted by simplependulum

evaluate the integral $\int_{0}^{\frac{\pi}{2}} x \cot(x) ~dx$

In fact , it is not a challenge problem for an experienced integrator
but i believe that this may help remind us of some techniques

let $x=u$ and $\cot x \ dx = dv.$ then integration by parts gives us: $I=\int_0^{\frac{\pi}{2}} x \cot x \ dx = -\int_0^{\frac{\pi}{2}} \ln(\sin x) \ dx.$ thus:
$-I=\frac{\pi}{2} \ln 2 + \int_0^{\frac{\pi}{2}} \ln(\sin(x/2)) \ dx + \int_0^{\frac{\pi}{2}} \ln(\cos(x/2)) \ dx$

$=\frac{\pi}{2} \ln 2 + \int_0^{\frac{\pi}{2}} \ln(\sin(x/2)) \ dx + \int_{\frac{\pi}{2}}^{\pi} \ln(\sin(x/2)) \ dx$

$=\frac{\pi}{2} \ln 2 + \int_0^{\pi} \ln(\sin(x/2)) \ dx=\frac{\pi}{2} \ln 2 + 2\int_0^{\frac{\pi}{2}} \ln(\sin x) \ dx$

$=\frac{\pi}{2} \ln 2 - 2I.$ therefore: $I=\frac{\pi}{2}\ln 2.$

**simplependulum** · June 8th 2009, 02:03 AM

terrific !

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