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Calculus: Help and Review13 chapters | 148 lessons

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Lesson Transcript

Instructor:
*Gerald Lemay*

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we use the product rule and integration by parts to find the integral of xe^x. The natural extension of integration by parts leads to a reduction formula which elegantly extends the integration results.

Let's start by differentiating *x*e*x* using the **product rule**. According to this rule, we take the derivative of the first function multiplied with the second function and add the first function multiplied with the derivative of the second. This sounds more complicated than it is, so let's look at an example:

On the right-hand side, the derivative of *x* is 1 and the derivative of e*x* is e*x*.

To clean up the right-hand side:

Now, we integrate both sides:

On the left-hand side, the integral 'undoes' the derivative, so the integral of d(*x*e*x*) is *x*e*x*.

And, on the right-hand side, the integral of the sum is the sum of the integrals:

This next part is a re-ordering of the three terms. We are looking for the integral of *x*e*x* so we place this term on the left-hand side by itself.

(In the next section, we will refer to this equation as the **integration by parts** formula.)

The integral of e*x* is just e*x*.

The constant 'C' is appended to the answer because this is an **indefinite integral** with unspecified limits of integration. And we are done!

Of course, we could factor out the e*x*:

Now to check this answer, we differentiate and the result should be just *x*e*x*.

Once again, we use the product rule:

and then expand and simplify:

The answer checks out!

So we now know what the answer is. What we've actually done by integrating the expansion of the product rule is to derive the **integration by parts** formula:

The key is selecting a *u* and d*v* which will reduce the complexity of the problem by giving us a simpler integral to solve. A good choice is:

For *u* = *x*, d*u* = d*x*. For d*v* = e*x* d*x*, *v* = e*x*.

Substituting *u*, d*v*, *v* and d*u* into the integration by parts formula:

Now simplifying, we have the following:

This gives us the same answer that we found before.

Let's explore what would have happened had we made the other choice for *u* and d*v*. This will actually lead to an interesting result.

Instead of *u* = *x*, we choose *u* = e*x*. Also, instead of d*v* = e*x* d*x*, we choose d*v* = *x* d*x*.

Thus, we now have d*u* = e*x* d*x* and *v* = *x*2/2. Summarizing this other choice:

Note the left-hand side of the integration by parts formula is the same, but the right-hand side has an integral, which has increased in complexity.

But the integral on the left-hand side is our friend e*x* (*x* - 1):

Thus, we can multiply by 2 and isolate the more complicated integral on the left-hand side:

Writing an integral in terms of another integral leads to the idea of a **reduction formula**.

There's a lot more that can be done.

Instead of integrating *x*1 e*x* or integrating *x*2 e*x*, we will integrate *x*n e*x*:

The integration by parts variables are:

Substituting into the integration by parts formula, we have the following:

For the integration on the right-hand side, pull out the *n* and re-order the terms:

We have two integrals. The one on the left-hand side, we will define as B*n* (*x*):

If *n* becomes *n* - 1, we get B*n*-1 (*x*):

which is the integral on the right of our integration by parts formula.

Thus:

This type of formula is called a reduction formula.

To use it, let *n* = 0. Then:

Now, for *n* = 1:

Do you see our integration of *x*e*x* result?

To get the integration of *x*2 e*x* result, let *n* = 2:

And we could continue with this reduction formula and keep getting new integral results based on previous results. How cool is that?

One way to find the integral of *x*e*x* is to use the **product rule** and then integrate. This is how the **integration by parts** formula is derived. In this lesson, we have **indefinite integrals** where the integration limits are not specified and, thus, a constant of integration is added to the result. Extending the idea of integration by parts leads naturally to a **reduction formula**, where an integral is defined in terms of a previously determined integral.

As you review the lesson, you may set the following goals:

- Use the product rule to solve for the integral of
*x*e*x* - Understand how to use the integration by parts
- Explain the use of indefinite integrals
- Detail the use of the reduction formula

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Calculus: Help and Review13 chapters | 148 lessons

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