MYTUTOR SUBJECT ANSWERS

509 views

how to integrate by parts

If you aren't lucky enough to be told in the question, the first thing we have to do is know when we should you integration by parts.

Well, like most things in maths this should become second nature to you with enough practice but as a general rule of thumb if we can't see a way past the problem after just looking at it and integration by substitution isn't a viable option, then this is when we should think about using intergration by parts.

Typical problems where this technique is suitable are where we have two quite different functions multiplied together, where one of those functions is differentiable infinitely many times.

For example: integrate x*sin(3x) with respect to x

Our formula for integration by parts is as follows:

integral(V*(du/dx)) = U*V - integral(U*(dv/dx))

In order to use this we must decide what is V and what is (du/dx), in short we want to label V as the function in the multiplication that will disappear after being differentiated a number of times which in most cases is a simple polynomial, this is because to use the formula we must solve the integral, integral(U*(dv/dx)), in this case we pick V = x and (du/dx) = sin(3x)

Note: if we picked sin(3x) to be V, then we would need to solve integral(U*[-cos(3x)/3]), which doesn't simplify the problem since U will only increase in order, in this case U = x2/2

So using our formula and the values of V and (du/dx) determined earlier,     V = x and(du/dx)= sin(3x)

This gives us integral(x*sin(3x)) = -[x*cos(3x)]/3 - integral(-[cos(3x)]/3)) which has now been reduced to a simple integral which is equal to:

-[x*cos(3x)]/3 - [sin(3x)]/9 + C

If there are no bounds don't forget the +C 

Tim M. A Level Maths tutor

2 years ago

Answered by Tim, an A Level Maths tutor with MyTutor


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

270 SUBJECT SPECIALISTS

£20 /hr

Ioannis P.

Degree: Computer Science (Bachelors) - Warwick University

Subjects offered:Maths

Maths

“Maths and Computer Science are both my passion. Having tutored students in the past, they think that my methods seem very intuitive and natural; you will too.”

£22 /hr

Benedict C.

Degree: Natural Sciences (Bachelors) - Leeds University

Subjects offered:Maths, Physics

Maths
Physics

“I am studying Physical Natural Sciences at Leeds University. My sessions focus on understanding and making examples work for you.”

£20 /hr

James M.

Degree: PhD in Mathematics (Doctorate) - Durham University

Subjects offered:Maths, Further Mathematics + 1 more

Maths
Further Mathematics
.STEP.

“Hi, I'm James, a friendly and easy-going PhD mathematician (it's not a lie!) who is here to help you understand and be able to do maths yourself!”

About the author

£20 /hr

Tim M.

Degree: Mathematics(G100) (Bachelors) - Bristol University

Subjects offered:Maths

Maths

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Using Integration by Parts, find the indefinite integral of ln(x), and hence show that the integral of ln(x) between 2 and 4 is ln(a) - b where a and b are to be found

Differentiation basics: What is it?

How do I integrate log(x) or ln(x)?

Sketch, on a pair of axes, the curve with equation y = 6 - |3x+4| , indicating the coordinates where the curve crosses the axes, then solve the equation x = 6 - |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