Integration by Parts

From the product rule, we can obtain the following formula, which is very useful in integration:

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It is used when integrating the product of two expressions (a and b in the bottom formula). When using this formula to integrate, we say we are "integrating by parts".

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Sometimes you will have to integrate by parts twice (or possibly even more times) before you get an answer.

Example

Find ∫xe^-x dx

Integrating by parts (with v = x and du/dx = e^-x), we get:

-xe^-x - ∫-e^-x dx         (since ∫e^-x dx = -e^-x)

= -xe^-x - e^-x + constant

We can also sometimes use integration by parts when we want to integrate a function that cannot be split into the product of two things. The trick we use in such circumstances is to multiply by 1 and take du/dx = 1. 

Example

Find ∫ ln x dx

To integrate this, we use a trick, rewrite the integrand (the expression we are integrating) as 1.lnx . We then let v = ln x and du/dx = 1 .

Hence ∫ ln x dx = x ln x - ∫ x (1/x) dx

= x lnx - ∫ dx

= x lnx - x + constant