int cos(x) ln(sin(x)) dx Evaluate the integral

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.

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