# What is integration by parts and why does it work?

##### 2 Answers

Integration by parts is a method of calculating integrals (both definite and indefinite).

#### Explanation:

Integration by parts is based on a formula:

It is a method to change one integral to another which may be easier to calculate.

This method is based on the formula for the derivative of ptroduct of 2 functions:

Now if we integrate both sides of the equation we get:

As an example I will calculate

In the example we changed an integral of

It works because of the product rule for derivatives and because integration is the inverse of differentiation.

#### Explanation:

The product rule tells us that

Using differentials, this says that:

more conveniently written

Now, Integrate both sides:

.

So,

By the inverse relationship of differentiation and integration, we can simplify the left side to get:

And finally, subtracting

**Alternative Notations**

Rather than using the dufferential notation shown above, others prefer to use:

Still others write

(Or something like that. I've seen the last one rarely.)