Integration by Substitution A.K.A... The Reverse Chain Rule

Integration by substitution is just the reverse chain rule. If you learned your derivatives well, this technique of integration won't be a stretch for you.

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Let's say we want to find this integral:

integration by substitution 1

For the sake of clarity, we´ll write it like this:

integration by substitution 2

Now, what is the derivative of sinx? It is cosx. So, what we have inside the integral sign is a composite function and its derivative.

Here we'll use a very clever trick. We'll change variable. Let's invent a variable called u, that would be equal to sinx:

integration by substitution 3

The derivative of u is:

integration by substitution 4

If we substitute these two equations into the integral we get:

integration by substitution 5

At this stage, you may not be very comfortable with differentials, but we can "cancel" them out:

integration by substitution 6

So, now we have an integral we already know how to solve:

integration by substitution 7

Finally, we just need to substitute u with sinx:

integration by substitution 8

And that's all integration by substitution is about. Now, let's derive our answer to check it. This is something you can always do check your answers:

integration by substitution 9

By the chain rule:

integration by substitution 10

And this is the function we wanted to integrate!

What we did with that clever substitution was to use the chain rule in reverse.

We saw that the integral was probably a composite function. This is because, according to the chain rule, the derivative of a composite function is the product of the derivatives of the outer and inner functions.

And that's exactly what is inside our integral sign.

As a rule of thumb, whenever you see a function times its derivative, you may try to use integration by substitution.

Let's do some more examples so you get used to this technique.

Integration by Substitution Example 2

Let's find this integral:

integration by substitution 11

Now, if you remember your derivatives, you know that the derivative of lnx is 1 over x.

To make clear why we need this fact, we'll write the integral like this:

integration by substitution 12

So, again, what we have is a function and its derivative. How should we choose our u?

Remember that we will derive u. So, we should always choose u as the function whose derivative is inside the integral sign. That means:

integration by substitution 13

So we have:

integration by substitution 14

This is an integral we know:

integration by substitution 15

Substitution back the u:

integration by substitution 16

Integration by Substitution Example 3

Let's find the integral:

integration by substitution 17

Will our u-substitution work here? Where is the derivative of the function there?

If we look closely, we'll note that we do have a composite function there.

We have the function ax. And then sin(ax), which is a composite function.

In our first example, we had a composite function and the derivative of the "inner" function. This integral is similar:

integration by substitution 18

Can you see it? What is the derivative of x? It is one!

So, let's choose our u:

integration by substitution 19

Its derivative:

integration by substitution 20

If we divide both sides of this equation by a, we get:

integration by substitution 21

Now, we can substitute that 1 in our integral:

integration by substitution 22

And we get:

integration by substitution 23

This integral is easier to solve. What is the function whose derivative is sin(u)? It is -cos(u):

integration by substitution 24

So, finally:

integration by substitution 25

Example 4

Now we'll do a tricky problem:

integration by substitution 26

We can write this as:

integration by substitution 27

We, again, have a composite function and its derivative. The harder part here is knowing how to choose our u:

integration by substitution 28

So, we have:

integration by substitution 29

If you look at your table of integrals, you'll find that:

integration by substitution 30

This fact is easily proved using trigonometric substitution. So, finally we have:

integration by substitution 31


  • Integration by substitution can be considered the reverse chain rule.
  • You'll need to know your derivatives well.
  • Whenever you see a function times its derivative, you might try to use integration by substitution.
  • With practice it'll become easy to know how to choose your u.

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