MYTUTOR SUBJECT ANSWERS

168 views

Prove that the indefinite integral of I = int(exp(x).cos(x))dx is (1/2)exp(x).sin(x) + (1/2)exp(x).cos(x) + C

Starting with the initial integral of int(exp(x).cos(x))dx we can see that this is going to have to be integrated by parts. This states that the integral of (u . dv/dx)dx is equal to u.v - int(v . du/dx)dx

Therefore, by applying this equation we can determine that u=exp(x), dv=sin(x), du=exp(x), v=-cox(x), as integrating sin(x) will give us -cos(x)

This gives us int(I) = exp(x).sin(x) - int(exp(x).sin(x))dx

As can be seen, this changes the form of the equation but it hasnt become any simpler. At this point we integrate once more by parts.

By looking at the 'int(exp(x).sin(x))dx' which we obtained, this can be integrated again.

int(exp(x).sin(x))dx = -exp(x).cos(x) + int(exp(x).cos(x))dx

Substituting this into the first integral we worked out will give us:

I = exp(x).sin(x) + exp(x).cos(x) - int(exp(x).cos(x))

It may seem that we have once again achieved nothing, but by inspecting the equation closely, we can see that we have ended up with the initial integral we were presented with on the RHS of the equation. By moving this negative integral to the other side we can see that we are going to have 2I. Dividing the whole equation by 2 will give us I = (exp(x).sin(x) + exp(x).cos(x))/2 + C (dont forget the constant!).

Hence we have obtained an answer to this cyclic integral.

Sammy A. A Level Maths tutor, GCSE Maths tutor, GCSE Chemistry tutor,...

7 months ago

Answered by Sammy, an A Level Maths tutor with MyTutor

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

179 SUBJECT SPECIALISTS

£20 /hr

Seb G.

Degree: Mathematics (Masters) - Bath University

Subjects offered: Maths

Maths

“Me: I'm a second year maths student at the University of Bath. I fell in love with maths after a rocky start with the subject, so believe that I can help people regardless of whether they like maths or not.  I've been tutoring maths l...”

MyTutor guarantee

£20 /hr

Jennifer L.

Degree: Mathematics with Finance (Bachelors) - Exeter University

Subjects offered: Maths, Chemistry

Maths
Chemistry

“Top tutor from Exeter University studying Maths with Finance, ready to help you improve your grades in Maths and Chemistry!”

MyTutor guarantee

£20 /hr

Linden S.

Degree: Economics and Finance (Bachelors) - Exeter University

Subjects offered: Maths, Economics

Maths
Economics

“I am currently in my third year studying economics and finance at the  University of Exeter. As well as having a strong passion for economics, I have a great love for mathematics too. I have experience in teaching through coaching ten...”

About the author

£20 /hr

Sammy A.

Degree: Chemical Engineering (Masters) - Birmingham University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
Chemistry

“Current first year student studying Chemical Engineering at the University of Birmingham. Have a love for solving all those tricky problems! If any help is needed in the realms of Physics/Maths/Chemistry I'd be happy to help and share ...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Prove that the d(tan(x))/dx is equal to sec^2(x).

Differentiate The Following function

How do you integrate e^x cos x

differentiate y=(4x^3)-5/x^2

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