How do you find the antiderivative of #cos(x) Ln(sin(x)) #?

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Answer 1 · 2017-08-14T12:55:51

Use substitution and parts.

#I = int cosxln(sinx) dx#

Let #t = sinx# so that #dt = cosx dx# and

#I = int lnt dt#

This can be integrated using parts

# u = lnt# so #du = 1/t dt#
#dv = dt#, so #v = t#

#I = tlnt-int t * 1/t dt#

# = tlnt- int dt#

# = tlnt-t+C#

Reverse the substitution to finish with

#int cosxln(sinx) dx = sinxln(sinx) - sinx+C#