Calculus

Integration of Functions

Integration of Functions Logo

Integration by Parts

Integration by Parts (IBP) is a special method for integrating products of functions. For example, the following integrals

in which the integrand is the product of two functions can be solved using integration by parts.

This method is based on the product rule for differentiation.

Suppose that u (x) and v (x) are differentiable functions. Then the product rule in terms of differentials gives us:

Rearranging this rule, we can write

Integrating both sides with respect to x results in

This is the integration by parts formula. The goal when using this formula is to replace one integral (on the left) with another (on the right), which can be easier to evaluate.

The key thing in integration by parts is to choose u and dv correctly.

The acronym ILATE is good for picking u. ILATE stands for

The ILATE acronym for integration by parts

The closer a function is to the top, the more likely that it should be used as

Solved Problems

Example 1.

Compute

Solution.

We use integration by parts:

Let Then

Hence, the integral is

Example 2.

Integrate by parts.

Solution.

We can choose because is simpler. Then

so we can easily integrate it and find the function

Apply the integration by parts formula:

The last integral is well known:

Hence

Example 3.

Evaluate the integral

Solution.

We choose

Hence

Substituting these expressions into the integration by parts formula

we have

Example 4.

Integrate

Solution.

We are to integrate by parts: The only choices we have for and are Then

Example 5.

Evaluate the integral

Solution.

Using the ILATE rule, we can choose

Then

Integrating by parts, we obtain

Example 6.

Evaluate the integral

Solution.

To use integration by parts we rewrite the integral as follows:

Now we can apply the ILATE rule, that is

This yields

Integrating by parts, we have

Example 7.

Compute the integral

Solution.

Keeping in mind the ILATE rule, we can choose

Then

This yields:

Example 8.

Evaluate the integral

Solution.

We set

Then

Using the integration by parts formula

we have

See more problems on Page 2.

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