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 differentiation 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.

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:

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:

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

Now, arranging this equality:

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:

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

Let's see an example.

Let's find the integral:

According to our formula:

So, here we have all set:

We can make:

So, we have:

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.

You can watch the video for this section:

Let's find:

We have our formula:

So, we choose:

And our integral becomes:

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

As always we must add the constant of integration.

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

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

For example:

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

So we have:

And our integral is:

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

And this is an easy integral:

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:

Here we'll choose:

And we have:

Taking the 2 out of the integral sign:

In example 1, we've found that:

We can replace this result to get:

And finally:

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

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:

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.

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:

Here we won't apply LIATE and make:

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

That is:

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

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

Our formula says:

Applying it in our integral:

And then:

And now we have something interesting. Look at this:

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:

Dividing both sides by two:

And that's the integral we were looking for!

If you've found this lesson useful, make sure you sign up to receive my **Free Intuitive Calculus Course**, if you haven't done so yet. If you're confused by this lesson, also make sure you subscribe, because the course will clear the doubts you have.

This course will help you bring home all the concepts we've covered in this page, plus the prerequisites and what comes after it.

This is a 10-Part course that contains the essential ideas, concepts and problem solving techniques of Calculus. Everything presented using my unique approach.

So, subscribe now if you haven't done so yet. Do this by using the sign up button below.

If you have just a general doubt about a concept, I'll try to help you. If you have a problem, or set of problems you can't solve, please send me your attempt of a solution along with your question. These will appear on a new page on the site, along with my answer, so everyone can benefit from it.

Click below to see contributions from other visitors to this page...

**Deduce General Formula Using Integration by Parts**

This is a very good exercise for practicing using a recurrence formula and also shows the generality of integration by parts . The problem is to find …

Return from **Integration by Parts** to **Integrals**

Return to **Home Page**

If you've found this page useful, you will love **Intuitive Calculus' Free eCourse. ***Since I started offering this free course, more than 5000 people have benefited from its intuition-first method, with special focus on problem-solving techniques.*

*This is a serious 10-part calculus course. The lessons provide links to external resources and videos whenever appropriate. A new lesson will be delivered to your e-mail inbox every day during the duration of the course. *

*You can unsubscribe at any time, although I doubt if you'll want to once you realize the true value of the course.*

*Your first lesson will be delivered to your inbox instantly after you sign up. *

## New! Comments

Do you have a doubt, or want some help with a problem? Leave a comment in the box below.