Substitution rule

Obviously, being able to find the antiderivative of a function is important, but the anti-differentiation formulas do not tell us how to evaluate every type of integral—for example, what to do when we have functions such as the following one:

This isn't as straightforward as the examples we saw earlier. In this case, we need to introduce a new variable to help us out and make the problem more manageable. 

Let's make our new variable u, and , and the differential of u is then . This changes the problem into the following:

This is clearly a lot simpler. The antiderivative of this becomes the following:

And by plugging in the original value , we get the following:

And there we have it.

This method is very useful, and works when we have problems that can be written in the following form:

If , then the following applies:

That equation might be looking somewhat similar to you. And it should. It is the chain rule from differentiation.