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

Expert Answers

An illustration of the letter 'A' in a speech bubbles

 

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.`

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial