MYTUTOR SUBJECT ANSWERS

501 views

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...

8 months ago

Answered by Darshan, an A Level Maths tutor with MyTutor

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

185 SUBJECT SPECIALISTS

£22 /hr

Hayden M.

Degree: Physics (Masters) - Warwick University

Subjects offered: Maths, Science+ 4 more

Maths
Science
Physics
Further Mathematics
English
Chemistry

“My name is Hayden. I am a 2nd year Physics student at The University of Warwick. I get excited aboutall things science and I hope that I can help you not only learn your material but alsounderstand why it is a useful thing to know.A...”

£20 /hr

Andrew B.

Degree: Economics L100 (Bachelors) - Exeter University

Subjects offered: Maths, Economics

Maths
Economics

“About Me: I am an economics student at Exeter University, I have a real passion for economics and how applicable it is to the world around us, and I hope my enthusiasm for it comes across in my tutorials and helps to inspire you! I'm...”

£30 /hr

Henri F.

Degree: Aerospace Engineering MEng (Masters) - Bristol University

Subjects offered: Maths, Physics+ 4 more

Maths
Physics
Further Mathematics
Electronics
.STEP.
.PAT.

“About Me I am an Aerospace Engineer currently studying my MEng at the University of Bristol. I have always been fascinated with space, spacecraft and satellites. I intend on going on to complete a PhD specialising inSpacecraft Control...”

About the author

Darshan S.

Currently unavailable: for regular students

Degree: Mathematics & Statistics (Bachelors) - Warwick University

Subjects offered: Maths

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

What is the quotient rule and how is it applied?

How and when do you use integration by parts?

What is 'Chain Rule' and why is it useful?

How do I find the equation of the normal line given a point on the curve?

View A Level Maths tutors

Cookies:

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

mtw:mercury1:status:ok