How do you evaluate the integral int sinthetaln(costheta)?

Jan 3, 2017

$\int \sin \theta \ln \left(\cos \theta\right) d \theta = - \cos \theta \left[\ln \left(\cos \theta\right) - 1\right] + C$

Explanation:

If we substitute:

$t = \cos \theta$
$\mathrm{dt} = - \sin \theta d \theta$

we have that:

$\int \sin \theta \ln \left(\cos \theta\right) d \theta = - \int \ln t \mathrm{dt}$

We can calculate this integral by parts:

$\int \ln t \mathrm{dt} = t \ln t - \int t d \left(\ln t\right) = t \ln t - \int \mathrm{dt} = t \ln t - t + C$

So, substituting back $\theta$ we have:

$\int \sin \theta \ln \left(\cos \theta\right) d \theta = - \cos \theta \left(\ln \left(\cos \theta\right) - 1\right) + C$