
We review common techniques to compute indefinite and definite integrals.

In some sense, calculating derivatives is straightforward. Most of the functions of interest are sums, scalar multiples, products, quotients, or compositions of common functions whose derivatives we can find. A “brute force” application of the sum, scalar multiple, product, quotient, and chain rules will eventually lead to a correct derivative. Integration is more of an art form. We often have to apply various techniques in order to write down a nice formula for the antiderivatives. We explore some of the common ones here.

Preliminary algebraic simplification

A good first step in attempting to compute antiderivatives involves simplifying the integrand first. Doing so requires careful algebra.

Substitution

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.

In the last example, after letting $u$ be the exponent and transforming $\d x$, there were no $x$ terms left. This does not always happen, but it does not mean that substitution does not aid in computing the indefinite integral.

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.

Splitting Up Fractions

In a similar vein, if you have a rational function, it can help to separate the integrand into several fractions.

Final Thoughts

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.