MYTUTOR SUBJECT ANSWERS

728 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...

9 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

183 SUBJECT SPECIALISTS

£20 /hr

Viktoriya L.

Degree: accounting and finance (Bachelors) - LSE University

Subjects offered: Maths, Russian+ 5 more

Maths
Russian
Mandarin
History
Economics
Accounting
-Personal Statements-

“Hello, My name is Viktoriya. I am a first year student at the London school of Economics and Poliical science studying accounting and finance. I am passionate about maths and finance in particular and wish to convey this passion to yo...”

£30 /hr

Louis S.

Degree: Mathematics (Bachelors) - Cambridge University

Subjects offered: Maths, Physics+ 6 more

Maths
Physics
Further Mathematics
Extended Project Qualification
.STEP.
.MAT.
-Personal Statements-
-Oxbridge Preparation-

“ second year undergraduate at the University of Cambridge, studying for a B.A. in Mathematics, having received A*s in A-Level Mathematics, Further Mathematics (Edexcel) and Physics (AQA)”

£22 /hr

Shivali J.

Degree: Medicine (Bachelors) - Imperial College London University

Subjects offered: Maths, Chemistry+ 5 more

Maths
Chemistry
Biology
.UKCAT.
.BMAT (BioMedical Admissions)
-Personal Statements-
-Medical School Preparation-

“Who am I? I am a first year medical student at Imperial College London. I have always been curious about the world we live in and have satisified this innate drive to understand the world around us through science. I hope to share thi...”

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

How does one find the derivative of ln(x)?

What is the importance of the Central Limit Theorem in carrying out hypothesis tests?

I did all the past papers but I still only achieved a C grade, what am I doing wrong?

The first term of an infinite geometric series is 48. The ratio of the series is 0.6. (a) Find the third term of the series. (b) Find the sum to infinity. (c) The nth term of the series is u_n. Find the value of the sum from n=4 to infinity of u_n.

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