Integration by Parts

Integration by parts is a method of integration that we use to integrate the product (usually !) of two functions. The aim is to change this product into another one that is easier to integrate. Although the formula looks quite odd at first glance, the technique is not too complicated.

\(\int { u \cdot\frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)


The tricky part is

1) deciding when we use it (the question will never say ‘solve this integral by the method of integration by parts’) and

2) deciding which part of the question we call u and which part \(\frac { dv }{ dx }\).

If you work through the resources on this page, you should be able to master these two things.


Key Concepts

On this page, you should learn about

  • integration by parts
  • repeated integration by parts

Essentials

The following videos will help you understand all the concepts from this page

What and Why

The formula for integration by parts is found from the Product Rule for differentiation. Here's how

Integration by parts is used for integrating the product of two functions (there are one or two exceptions to this that we will see later). We use this technique when Integration by Substitution (or recognition) does not work. You can see in the table below some examples of when we use each technique

Integration by Substitution Integration by Parts
\(\int { 2x\cdot { e }^{ -3{ x }^{ 2 } }dx } \) \(\int { 2x\cdot { e }^{ -3{ x } }dx } \)
\(\int { sinx\cdot { e }^{ cosx }dx } \) \(\int { sinx\cdot { e }^{ x }dx } \)
\(\int { x\cdot \sqrt { x+1 } dx } \) \(\int { x\cdot \sqrt { x+1 } dx } \)

Notice that it is sometimes possible to use either technique, as is the case in the last example!

Deciding on u and dv/dx 1

When using the formula for integration by parts you need to learn which part of the product you are going to call u and which part \(\frac { dv }{ dx }\).

\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)

Generally, you should

a) Let be the function that gets simpler when you differentiate

b) Let \(\frac { dv }{ dx }\)be the function that doesn't get too complicated when you integrate

Here is a video that will help you with that. It shows what happens when you make the correct choice and what happens when you make the wrong choice.

Use integration by parts to find \(\int { 2x\cdot { e }^{ -3{ x }}dx } \)

Deciding on u and dv/dx 2

There following list might help you with your choice. Look at the two functions in your question. The one that appears first on the following list is the one you should choose to be u

  1. Logs
  2. Inverse Trig
  3. Algebra
  4. Trig / Exponential

Here's another example

Use integration by parts to find \(\int { x^ 2\cdot lnxdx } \)

Repeating the Process 1

Sometimes you have to apply the method of integration by parts more than once!

Use integration by parts to find \(\int { x^{ 2 }\cdot sin\left( \frac { x }{ 2 } \right) dx } \)

Repeating the Process 2

Sometimes integration by parts really doesn't seem to help and we end up going round in circles! Here's an example where that is the case and a clever solution to it!

Use integration by parts to find \(\int { e^{ x }\cdot cos2xdx } \)

Special Integrals

Integration by parts is mainly used for integrating the product of two functions. However, we can use it to find the integral of ln(x) and arctrigonometric functions too. The second question requires integration by substitution as well as integration by parts!

Use integration by parts to find \(\int { lnx } dx\) and \(\int { arcsinx } dx\)

Summary

Print from here

Test Yourself

This is the formula for integration by parts.

\(\int { u \cdot \frac { dv }{ dx } dx } \quad =\quad uv\quad -\quad \int { v\cdot \frac { du }{ dx } } dx\)

Here is a quiz that gets you to practice making the correct choice for u in the  formula


START QUIZ!  

Here is a quiz that gets you to practise the method of integration by parts


START QUIZ!

Exam-style Questions

Question 1

\(\int { { e }^{ \frac { x }{ 2 } }sin(x)\quad dx } \)

Hint

Full Solution

 

Video Solution

Question 2

\(\int { arctanx \ dx } \)

Hint

Full Solution

 

Video Solution

Question 3

\(\int { 2x\cdot arctanx \ dx } \)

Hint

Full Solution

 

Video Solution

MY PROGRESS

How much of Integration by Parts have you understood?