There are a number of integrals that we can evaluate directly, called elementary functions:

integral[*x dx*] = *x*^{2}/2 + C

integral[cos(*x*) *dx*] = sin(*x*) + C

However, most integrals cannot be solved so directly. To illustrate the technique of integration by substitution, let’s look at an example.

Example: Find I, where

I = integral[*x*cos(*x*^{2}) *dx*]

It’s not immediately obvious how to start here. But, if you look closely, you can see that there is a part of the function (*x*) which is the derivative of another part (*x*^{2}). This is the key to the problem!

The form of the integral should remind you of the chain rule. Integration by substitution is a way of “reversing” the chain rule in a more formal manner.

The fundamental step here is to change the variable of the function in order to simplify it. If we make the substitution *u* = *x*^{2}, then we can see that this would make the integral a lot simpler. This is because we know how to integrate cos(*u*) directly.

However, if we change the variable in this way, we will end up integrating *u* with respect to *x* (remember, that’s what the *dx* bit of the integral means!). Clearly this doesn’t make sense. To get round this problem, we have to integrate with respect to *u*. Fortunately, we can find this out by differentiating both sides of the substitution:

*du*/*dx* = 2*x*

Now, a word of warning. At this level, we can assume that it is safe to treat *du*/*dx* as a fraction, and rearrange it to find *dx* as the subject. Mathematically speaking, however, this is not the case. So you should be aware that this is simply a method; if you were studying maths at a higher level, you would find a rigorous argument to show that you can perform this calculation.

With that aside, let’s rearrange the expression like so:

*dx* = *du*/2*x*

We are finally ready to put all the substitutions in:

I = integral[*x*cos(*x*^{2}) *dx*] = integral[*x*cos(*u*) *du*/2*x*] = (1/2)integral[cos(*u*) *du*]

(Note: you can take constant factors outside of the integral sign!)

Hopefully you can now see the point of the substitution, which probably seemed a bit strange to begin with! Let’s solve the integral now:

I = (1/2)integral[cos(*u*) *du*] = (1/2)sin(*u*) + C

We’re not quite done yet. Our original integral was in terms of *x*, but our answer is still in terms of *u*. We need to change the variable back to *x*; fortunately, this is easy to do:

I = (1/2)sin(*x*^{2}) + C

__Summary__

Integration by substitution should be used if the integral looks as though it came from differentiating a function of a function (the chain rule).

We change the variable to *u* using the appropriate substitution.

We need to remember to find *dx* in terms of *du*.

If we are finding a definite integral, we also need to change limits.

Hopefully, this should lead to a solvable integral!

Remember to change the variable back if we are finding an indefinite integral; if we are finding a definite integral, we can just substitute in the changed limits.