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#