MYTUTOR SUBJECT ANSWERS

657 views

How to integrate by parts

Sometimes while integrating, we may come across an expression that is not a polynomial, and thus we cannot use the convenient power rule to integrate. Consider the function y=xcos(x). It is not immediately clear how we should start we this one, however upon further inspection, we may introduce the technique of Integrating by Parts. Essentially we split the function into two parts, say u and v, and then employ a formula which allows us to integrate them together:

∫u·dv = u·v − ∫v·du

Applying this to our function, we obtain

∫x·cos(x) dx = u·v − ∫v·du

Here we note that integrating cos(x) is a lot simpler than integrating x, and differentiating x is also simpler than differentiating cos(x), so it would make sense to set

u = x and dv = cos(x)

This in turn gives us

du = 1 and v = sin(x)

Thus plugging these values back into our original formula, we get

∫x·cos(x)= x·sin(x) - ∫sin(x)·1

So now, all we need to do is integrate sin(x), which is definitely easier than what we started with. Thus, the end product gives us 

∫x·cos(x)= x·sin(x) + cos(x) + C

where C of course is the constant of integration.

Dennis L. IB Maths tutor, A Level Maths tutor, GCSE Maths tutor, GCSE...

1 year ago

Answered by Dennis, an A Level Maths tutor with MyTutor


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

310 SUBJECT SPECIALISTS

£24 /hr

Ayyub A.

Degree: Mathematics Msci (Bachelors) - Bristol University

Subjects offered:Maths, Further Mathematics + 1 more

Maths
Further Mathematics
.MAT.

“Hi I'm Ayyub, current student at Bristol university studying Maths. I have significant tutoring experience in maths and hopefully i'll be able to help you out.”

£24 /hr

Sam F.

Degree: Economics with Placement (Bachelors) - Bath University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Extended Project Qualification
Economics

“Studying for BSc Hons Economics, A level economics, maths and physics. Able to tutor GCSE/AS/A2 Economics, Maths and GCSE physics! ”

£20 /hr

Eleanor J.

Degree: Natural Sciences in Maths and Physics (Masters) - Durham University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“Study Maths and Physics at Durham Previous tutoring experience Enjoy helping students improve and enjoy the subjects Adaptable to different academic needs”

About the author

Dennis L.

Currently unavailable: for regular students

Degree: Mathematics (Bachelors) - Edinburgh University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Chemistry

“Hi!I'm currently an undergraduate student studying Mathematics at the University of Edinburgh. I have always found mathematics to be the most interesting and thought-provoking, which is why I have chosen to study it at university and...”

You may also like...

Other A Level Maths questions

The polynomial x^3 - 2*x + a, where x is a constant is denoted by p(x). It is given that x+2 is a factor of p(x). Find a

The curve C has equation y = x^3 - 2x^2 - x + 9, x > 0. The point P has coordinates (2, 7). Show that P lies on C.

What is the derivative of ln(x)?

How do you differentiate y = 5 x^3 + 1/2 x^2 + 3x -4

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