We learn a new technique, called substitution, to help us solve problems involving integration.

Computing antiderivatives is not as easy as computing derivatives. One issue is that the chain rule can be difficult to “undo.” We have a general method called “integration by substitution” that will somewhat help with this difficulty.

If the functions are differentiable, we can apply the chain rule and obtain If the derivatives are continuous, we can use this equality to evaluate a definite integral. Namely,

On the other hand, it is also true that

Since the right hand sides of these equalities are equal, the left hand sides must be equal, too. So, it follows that

This simple observation leads to the following theorem.

The integral on the right appears to be much simpler than the original integral. So, when evaluating a difficult integral, we try to apply this theorem and replace the original integral with a simpler one. We can do this as long as the conditions of the theorem are satisfied.

We will apply the Integral Substitution Formula (ISF) to the following example.

We can solve a problem like this in a slightly different way. Let’s do the same example again, this time we will think in terms of differentials.

Finally, sometimes we simply want to deal with the antiderivative on its own. We will repeat the example one more time in oder to demonstrate this approach.

More examples

With some experience, it is (usually) not too hard to see what to substitute as . We will work through the following examples in the same way that we did for Example ??.

If substitution works to solve an integral (and that is not always the case!), a common trick to find what to substitute for is to locate the “ugly” part of the function being integrated. We then substitute for the “inside” of this ugly part. While this technique is certainly not rigorous, it can prove to be very helpful. This is especially true for students new to the technique of substitution. The next two problems are really good examples of this philosophy.

To summarize, if we suspect that a given function is the derivative of another via the chain rule, we introduce a new variable , where is a likely candidate for the inner function. We rewrite the integral entirely in terms of , with no remaining in the expression. If we can integrate this new function of , then the antiderivative of the original function is obtained by replacing by .