Here we are going to come up with our second method for solving integrals which we can’t solve by inspection (noting that they are the antiderivative of some function which we know well).

We know the product rule for differentiation:

Integrating it gives us:

We can then rearrange this to give:

We have dropped the because this will come also from doing the second integral.

*This is also sometimes written in an alternative form:*

Let and . Sometimes we will keep the functional dependence explicit, and sometimes we won’t – it is up to you to follow what things are functions of and what are constants. You will get more and more familiar with it over time.

Now we can write:

Now put this into the equation above for integration by parts.

cancelling the factors of :

(Actually the idea of cancelling the factors of is rather sloppy, but here it goes through ok).

Whenever we are doing an integration by parts our job is to figure out which function we are going to choose to be and which part is going to be .

### Some examples

we will generally choose (ie. ) to be the function which when differentiated becomes simpler. In this case we choose and . Then and .

Plug this into the formula:

Let’s try a simple example:

If we take and , then and . Then the equation becomes:

**A general rule, but not to be taken as sacred**: Very often you can use integration by parts if you have a product of terms, one of which will simplify when you integrate it, and one of which will simplify when you differentiate it. The one that simplifies when you differentiate it should be , and the one that simplifies when you integrate (multiplied by ) should be .

mackkJuly 29, 2015 at 7:10 amJon i get 1/2x^2lnx-1/2x^2 if i swap u and dv

Jonathan ShockJuly 29, 2015 at 7:31 amHi Mackk, do you want to pop by during office hours and we can talk through this example. You certainly should be getting the same answer…

All the best,

J

Mathemafrica - UCT MAM1000 lecture notes subject listOctober 4, 2015 at 10:27 am[…] MAM1000 lecture notes part 1, part iii – Integration by parts […]

Jess BournJuly 19, 2016 at 6:43 amThis is so useful and really well explained. It’s very helpful to be able to see these Mathemafrica posts before coming to the lectures as it makes the lecture so much easier to follow. Would it be possible to make these pre-lecture posts a regular thing? Even if not for all lectures, just the ones that you think might be the harder ones to follow? It makes learning a lot easier!

Jonathan ShockJuly 19, 2016 at 6:47 amHi Jess, that’s the plan