Integration by parts

By now, we know that for every rule in differentiation, there is a corresponding rule in integration since they have an inverse relationship.

In the earlier section on differentiation, we encountered the product rule. In integration, the corresponding rule is known as integration by parts.

As a recap, the product rule states that if f and g are differentiable, then the following applies:

And so, in integration, this becomes the following:

We can rewrite this, as follows:

We can combine this formula with the fundamental theorem of calculus and obtain the following equation:

We can use this to evaluate the integral between the interval [a, b]. 

Note: The term   merely states that we plug in the value b in place of x and evaluate it, and then subtract it from the evaluation at a.

We can also use the preceding substitution method for integration by parts to make our lives easier when calculating the integral. We make  and ; then, the differentials are  and . And so, the formula becomes this: