# Evaluate the indefinite integral.   integrate of cot(x)ln(sin(x))dx

Luca B. | Certified Educator

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You should evaluate the integral using the following substitution such that:

sin x = t => cos x dx = dt

Changing the variable yields:

int ln(sin x) cos x dx = int ln t dt

You should use integration by parts formula such that:

int udv = uv - int vdu

Reasoning by analogy yields:

u = ln t => du = (dt)/t

dv = dt => v =  t

int ln t dt = t*ln t - int t*(1/t)dt

int ln t dt = t*ln t - int dt => int ln t dt = t*ln t - t + c

Factoring out t yields:

int ln t dt = t*(ln t - 1) + c

Substituting back sin x  for t yields:

int ln(sin x) cos x dx = sin x*(ln(sin x) - 1) + c

Hence, evaluating the given indefinite integral , using two methods of integration, yields int ln(sin x) cos x dx = sin x*(ln(sin x) - 1) + c.

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jeew-m | Certified Educator

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Let;

t = lnsinx

Then;

(dt)/dx = 1/sinx*cosx = cotx

dt = cotxdx

int cotxlnsinxdx

= int lnsinxcotxdx

= int tdt

= t^2/2+C where C is a constant

= 1/2(lnsinx)^2+C

= lnsinx+C

int cotxlnsinxdx = lnsinx+C

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