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

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Answer 1 · 2017-01-03T14:52:06

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

If we substitute:

$t=cos theta$
$dt = -sin theta d theta$

we have that:

$int sin theta ln (cos theta) d theta = -int lnt dt$

We can calculate this integral by parts:

$int lnt dt = tlnt -int t d(lnt) =tlnt - int dt = tlnt-t +C$

So, substituting back $theta$ we have:

$int sin theta ln (cos theta) d theta = -costheta(ln(cos theta) -1) +C$

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