Integration by Parts

Animated Banner

Integration by parts is a calculus technique, which allows mathematicians to integrate large functions by separating them out into parts.

General Formula

The basic formula equation shown at the top of this page is the one to use for solving integration by parts problems. The product rule used in differentiation derives this formula, and you can see exactly how in the following sections of this multi-page article.

How to Choose u

Choosing the order of u is a simple process and takes a little time to build intuition. In fact, if you are just starting out, then making a mistake is the best thing because it gives you an insight into why it is so important. If you find that one of your terms keeps repeating and you are going in circles, then you can be sure that the substitution may not be in the correct order because the expression does not reduce. You can gain a lot of intuition when making a mistake like that.

Letter Stands for Example
I Inverse Trig arcsin, arccos, arctan ...
L Logarithmic ln x, Log ...
A Algebraic x, x^2, x^3, ax^n ...
T Trig sin x, cos x, tan x, cot x ...
E Exponential e^x, a^x, x^x ...

The order of u: This is the order in which to choose u. Therefore, you choose u first from the list and then dv/dx will be whatever that is left. Another way to think about this is when you choose for dv/dx, make sure it is something that you can integrate easily to find v.

When you choose for u, make sure it is something that you can differentiate easily to find du/dx. Ideally, you want the derivative to be something much simpler because you will need to integrate it later.

It is very difficult to pass on intuition on a web page, but the main things you should focus on are the following.

  • Is there a derivative?
  • Will it integrate?
  • Does it simplify?


The best way to learn is to work independently on the examples found through the links below this page. I would recommend going through each example starting with the simple ones such as the exponential.

Have the relevant page open in front of you and then on a piece of paper see if you can follow it. Try to figure out the steps on your own and only look at the screen when you are stuck. Each example has the standard formula in the header section so it should help.

My favourite example is the sin x e^x where you can see one of the terms keeps repeating. The solution to this type of problem, when you are going in circles, is interesting and something my teacher Mr Smith showed me back in the late 1980s.

Calculator Solver

Although you can get calculators that can do this type of algebra, I would highly recommend you learn to do this by hand using some brain cells. This is vital if you are planning to become an engineer.

If you learn to do the solution by hand then you will actually understand what is happening and apply the intuition to other more complex problems.

This Article Continues...

Integration by Parts
Integration by Parts Derivation
Integration by Parts xe^x
Integration by Parts xe^2x
Integrate xe^3x
Integration by Parts ln x
Integration by Parts xln x
Integrate xe^x^2
Integrate arcsin x
Integration by Parts arctan x
Integration by Parts Exponential
Integrate e^x sin x