MYTUTOR SUBJECT ANSWERS

485 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,...

1 year ago

Answered by Sammy, an A Level Maths tutor with MyTutor


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

308 SUBJECT SPECIALISTS

£24 /hr

Ayusha A.

Degree: BEng electrical and electronics engineering (Bachelors) - Newcastle University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“About me: I am a final year Electrical and Electronic Engineering student at Newcastle University. I took Mathematics, Further Mathematics, Chemistry and Physics as my A-level subjects. I did peer mentoring in university and also have...”

PremiumTimothy N. A Level Design & Technology tutor, GCSE Design & Technolog...
£36 /hr

Timothy N.

Degree: Architecture and Environmental Engineering (Masters) - Nottingham University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Design & Technology
-Personal Statements-

“Hi there, I have a passion for helping students achieve, and believe that with my years of experience tutoring, we will be able to surpass the grades you want!”

Ashika V. GCSE Maths tutor, A Level Maths tutor, IB Maths tutor, 13 P...
£26 /hr

Ashika V.

Degree: Physics (Masters) - Manchester University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Extended Project Qualification
-Personal Statements-

“Hi I'm Ashika, a third year MPhys Physics student at the University of Manchester. I tutor in Maths and Physics, Personal Statements and the EPQ.”

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

How do I differentiate y = ln(sin(3x))?

A line L is parallel to y = 4x+5 and passes through the point (-1,6). Find the equation of the line L in the form y = ax+b.

Consider the functions f and g where f (x) = 3x − 5 and g (x) = x − 2 . (a) Find the inverse function, f^−1 . (b) Given that g^−1(x) = x + 2 , find (g^−1 o f )(x) . (c) Given also that (f^−1 o g)(x) = (x + 3)/3 , solve (f^−1 o g)(x) = (g^−1 o f)(x)

By writing tan x as sin x cos x , use the quotient rule to show that d dx ðtan xÞ ¼ sec2 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

mtw:mercury1:status:ok