Integrate sinx*ln(cosx) with respect to x.

1. ʃ sinx*ln(cosx) dx

First notice the composition ln(cosx). To make the expression easier to integrate we try substitution (substitution is often useful when trying to integrate expressions that are or include compositions of functions).

2. So we let t=cosx.

Now we must find out what dx is in terms of dt. This is quite simple by just differentiating the substitution.

3. dt/dx=-sinx <=> dx=(-1/sinx)dt

Now we substitute everything into our integral from 1.

4. ʃ sinx*ln(t)*(-1/sinx) dt

The sinx from the top and bottom cancel.

5. ʃ -1*ln(t) dt

I have written a -1 since we will have to use integration by parts to integrate ln(t) with respect to t. (Reminder: ʃ u(dv/dx) dx = uv - ʃ v(du/dx) dx)

6. Let u=ln(t) and dv/dt=-1

Now we need to differentiate u to find du/dt and integrate dv/dt to find t.

7. du/dt=1/t and v=-t

Use the intergation by parts formula and that uv=-tln(t) and v(du/dt)=-t(1/t)=-1.

8. ʃ -1*ln(t) dt = -tln(t) - ʃ -1 dt = -tln(t)+t+C

! Remember the constant of integration and that  because we used a substitution we must give our answer back in terms of x.

9. So using 2. again we get cosx(1-ln(cosx))+C as the answer.

Darshan S. A Level Maths tutor, 13 plus  Maths tutor, 11 Plus Maths t...

1 year ago

Answered by Darshan, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Scott E.

Degree: Computer Science (Bachelors) - Warwick University

Subjects offered:Maths, Physics+ 1 more


“Maths and Computer Science lover. With a patient, thorough and friendly teaching style.”

£30 /hr

Anuradha D.

Degree: Physics (Masters) - Durham University

Subjects offered:Maths, Physics+ 4 more

Further Mathematics
-Personal Statements-

“Hello everyone! My name is Anuradha (you can call me Anu), and I am a Physics Undergrad at Durham. I have always had a “thing” for science and am so pleased to be studying it – and hope that I can give you all the support you need to ...”

PremiumGeorge B. GCSE Maths tutor, A Level Maths tutor, A Level Further Math...
£26 /hr

George B.

Degree: Mathematics (Masters) - Warwick University

Subjects offered:Maths, Further Mathematics + 1 more

Further Mathematics

“Premium tutor. First class graduate with teaching experience from a top Russell Group university. I deliver fun and relaxed lessons which achieve results!”

About the author

Darshan S.

Currently unavailable: for regular students

Degree: Mathematics & Statistics (Bachelors) - Warwick University

Subjects offered:Maths


“About Me:  I am currently studying at the University of Warwick in my first year reading Mathematics and Statistics.  Throughout secondary school, maths was always my favourite lesson not only because I excelled in it but also becaus...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Integrate (x)(e^x) with respect to x and then integrate (x)(e^x) with respect to y.

Showing all your working, evaluate ∫(21x^6 - e^2x- (1/x) +6)dx

Solve the equation " 2sec^2(x) = 5tanx " for 0 < x < π

Integrating cos^2(x)+5sin^2(x)

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss