Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 33

You need to use the substitution sin x = t, such that:
sin x = t => cos x dx = dt
Replacing the variable, yields:
int cos x*ln(sin x) dx = int ln t dt
You need to use the integration by parts such that:
int udv = uv - int vdu
u = ln t => du = (dt)/t
dv = 1 => v = t
int ln t dt = t*ln t - int t*(dt)/t
int ln t dt = t*ln t - int dt
int ln t dt = t*ln t - t + C
Replacing back the variable, yields:
int cos x*ln(sin x) dx = sin x*ln (sin x) -sin x+ C
Hence, evaluating the integral, using substitution, then integration by parts, yields int cos x*ln(sin x) dx = sin x*(ln (sin x) -1)+ C.