Integration By Parts


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For finding the integral of the product of two functions, you will need to use integration by parts.

The formula in function notation is:

Note: This product as seen in the formula is the product of u and dv/dx.

How do you know which function is u and which function is dv/dx?

It will come with practice, but usually for expressions which contain xn, the u term should be the xn term. One case where it is different is an expression like x2lnx. Here the lnx term should be the u term, because it is difficult to integrate to find v if dv/dx is the lnx term. Likewise, if there is an ex term in the integral, it should be dv/dx, as it is easy to integrate that kind of term to find v.

Example 1:

Example 1
Answer 1
Example 2
Answer 2

Example 2:

Example 1
Answer 1
Example 2
Answer 2

Sometimes, you may need to use the formula twice in a question.

Example 3:

Example 1
Answer 1
Example 2
Answer 2

The following practice question is a difficult problem, with a trick at the very end.

Practice Question:





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Note: Although the examples do not cover definite integrals, the process is the same to solve a normal definite integral. There is no need to convert limits, not like if you were integrating with substituition.