Integration by Parts
Intuition and Useful Tricks

Integration by parts is a "fancy" technique for solving integrals. It is usually the last resort when we are trying to solve an integral.

The idea it is based on is very simple: applying the product rule to solve integrals.

So, we are going to begin by recalling the product rule. Using the fact that integration reverses integration we'll arrive at a formula for integrals, called the integration by parts formula.

We'll then solve some examples also learn some tricks related to integration by parts.

The Integration by Parts Formula

In the following video I explain the idea that takes us to the formula, and then I solve one example that is also shown in the text below.

In the video I use a notation that is more common in textbooks. The video and the text below cover the same ideas, but because of that I recommend you to both watch the video and read.

Let's begin by recalling the product rule for derivatives:

The product rule

This formula is easy to memorize.

Now, what we have is an equality of two functions. If two functions are equal, their integrals, or antiderivatives, must also be equal.

Let's find the antiderivatives of both sides:

Integrating the product rule

As integration reverses differentiation, we have that the integral and the derivative on the left side "cancel each other":

Rearranging the product rule

Now, arranging this equality:

Integration by parts formula

Look at that. Using just the product rule we obtained an interesting formula for integration. This formula is called the integration by parts formula.

Many people use the letters u and v instead of f and g. In the videos I'll use these letters. 

When we use this formula, we "divide the integral in parts". We must find another integral, but this one is hopefully simpler.

Integration by parts is useful when the function we want to integrate can be written as:

Functions that can be integrated by parts

That is, the product of a function f and the derivative of another function g.

Let's see an example.

Example 1

Let's find the integral:

Example 1 of integration by parts

According to our formula:

Integration by parts formula

So, here we have all set:

Determining f(x) and g'(x) in integration by parts

We can make:

Finding the other terms in the integration by parts formula

So, we have:

Applying integration by parts

Easy enough. Now, you might ask a very important question: how did we choose our f(x) and g'(x)?

As a rule of thumb, you must choose f(x) such that its derivative is simpler than f(x).

Don't worry, the ability to choose f(x) correctly will be second nature to you after doing many exercises. In the previous example we had f(x) = x. Its derivative is one. That is much simpler!

In a later section I'll show you a very easy to remember rule for choosing f(x), called the LIATE rule.

Example 2

You can watch the video for this section:

Let's find:

Example 2 of integration by parts

We have our formula:

Applying the integration by parts formula

So, we choose:

Determining the terms in integration by parts

And our integral becomes:

Applying integration by parts

And the antiderivative of -cos(x) is -sin(x). So, our integral is:

Answer to example 2 of integration by parts

As always we must add the constant of integration.

Trick Nº 1

Integration by parts is very "tricky" by nature. Here I'll show you one special trick. In the formula:

Integration by parts formula

We can consider g'(x) = 1. This is useful because that function can always be written in an integral.

For example:

Example 3 of integration by parts

It is surprising we don't know this integral yet! We can write it like this:

Applying the integration by parts formula in a tricky way

So we have:

Determining the terms in the integration by parts formula

And our integral is:

Applying integration by parts

And then, simplifying the x's in the integral:

Applying integration by parts

And this is an easy integral:

Answer to example 3 of integration by parts

Trick Nº 2

This second trick could be considered more a method than a trick. Integration by parts is usually the last resort for solving an integral.

When you apply it you may still get an integral that you don't know how to solve. What do we do in those cases? Integrate by parts again! An example:

Example 5 of integration by parts: trick 1

Here we'll choose:

Determining the terms of integration by parts formula

And we have:

Applying integration by parts formula to example 5

Taking the 2 out of the integral sign:

Applying the integration by parts formula

In example 1, we've found that:

Applying integration by parts a second time

We can replace this result to get:

Answer to example 5 of integration by parts

And finally:

Answer to example 5 of integration by parts

If you find this integral in an exam, for example, you'll need to use the formula twice.

The Secret

When you first learn this technique of integration, the difficulty is in choosing f(x) and g'(x) correctly. What happens if you don't choose well? Your integral won't simplify.

Here's a rule of thumb I learned when studying integration, and I've been using it since then.

It was proposed by Herbert Kasube of Bradley University. It is called the LIATE rule. LIATE stands for:

L: Logarithmic functions

I: Inverse trigonometric functions

A: Algebraic functions: x squared, etc.

T: Trigonometric functions

E: Exponential functions

How do we use it? Whatever function from that list comes first should be our f(x).

For example, let's consider again the integral:

LIATE rule example

Thee we have the product of two functions: one algebraic (x) and one trigonometric.

According to LIATE, A (algebraic) comes before T (trigonometric). So, the algebraic function (x in this case) should be our f(x).

If you check all the examples we did you'll see tat we followed LIATE.

The next time you need to apply integration by parts try to use the LIATE rule, I'm sure you'll be surprised at how effective it is.

Trick Nº 3

Here we'll solve one of the few integrals that don't follow the LIATE rule. This one is a trick question in many exams:

Tricky integral by integration by parts

Here we won't apply LIATE and make:

Determining the terms in tricky integral by parts

You'll have to trust me on this one. We have:

Applying integration by parts formula for tricky integral

That is:

Result of applying integration by parts on tricky integral

Here we'll apply a trick we've already learned and use integration by parts again:

We need to apply integration by parts a second time

This time we call our functions u(x) and v'(x):

Determining the terms in the second application of integration by parts formuala

Our formula says:

Integration by parts formula

Applying it in our integral:

Applying integration by parts a second time

And then:

Algebraic expression of tricky integral

And now we have something interesting. Look at this:

Solving tricky integral algebraically

Here we have an equation. So, we can still apply the rules of algebra. We can add the same integral to both sides and get:

Answer to tricky integral by parts

Dividing both sides by two:

Answer to tricky integral by parts

And that's the integral we were looking for!


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