We review common techniques to compute indefinite and definite integrals.
A good first step in attempting to compute antiderivatives involves simplifying the integrand first. Doing so requires careful algebra.
While it may be tempting, note that . We can never simplify by cancelling terms over addition or subtraction in the denominator. Instead, we can split the fraction up then simplify.
Sometimes, it is helpful to try to recognize when an integral involves reversing the chain rule for differentiation. When an integrand involves a composition of a trigonometric, exponential, or power and another function, letting a new variable represent the inner function helps.
Note that no algebra will help us simplify the expression. The only antidifferentiation formula we have regarding the exponential is . Let’s start by letting the exponent in the original integral be a new variable :
Let . We now need to write everything in the integral in terms of this new variable. Recalling that if , the differentials are related by , we can write .
We thus find
The antiderivative in is now easy to compute.
Reversing the substitution by setting above now gives .
In the last example, after letting be the exponent and transforming , there were no terms left. This does not always happen, but it does not mean that substitution does not aid in computing the indefinite integral.
Note that we now have an extra term in the integrand. Since depends on , we cannot move it outside of the integral. Before abandoning hope that our substitution is useful here, we can ask whether it is easy to express this leftover in terms of .
Since , we find that , so we can now rewrite the integral as:
We can now perform some algebra, then integrate.
Substituting to obtain the antiderivatives in terms of , we find that
Let’s see another example involving a trickier substitution and a definite integral. First, recall the substitution theorem for definite integrals:
This may look difficult to read, but it really just reminds us that to work with definite integrals, we need to write the integrand, the differential, and the limits of integration in terms of the new variable.
For the limits of integration, note that when , we have . Similarly, when , .
Since , we find .
We can now write our original integral in terms of :
We can now evaluate the integral:
In a similar vein, if you have a rational function, it can help to separate the integrand into several fractions.
Understanding how to integrate is highly dependent on having a good grasp on how to differentiate functions and being comfortable with algebra. The techniques presented here are not an exhaustive list. In order to become proficient computing integrals, there is no substitute for practice.