Mastering Mathematics Smartly
by Wee Wen Shih

A unique self-help website that provides comprehensive coverage of mathematics at A-level & beyond, written in a student-friendly style.

Integrate by parts

Typically, an expression to be integrated has two parts, for example: x ln x.

You need to decide which part to integrate and which part to differentiate.

The "LIATE" rule helps. Each letter refers to a type of expression:

L ogarithmic expression
I nverse trigonometric expression
A lgebraic expression
T rigonometric expression
E xponential expression

For the expression x ln x, x falls under 'A' and ln x falls under 'L'.

'L' comes before 'A', so we choose to differentiate ln x and integrate x.

See example.

Try this!

Exceptions to the rule

Sometimes, you may face a situation where there is a single term in the given expression to be integrated. For example, consider the integral \int_{}^{} \text{ln} x\:dx. The approach is to introduce a '1' to make the expression contain two terms, i.e. \int_{}^{} 1.\text{ln} x\:dx.

On some other occasions, you may face a situation where both terms belong to the same type. For example, consider the integral \int_{}^{} (\text{tan} x)(\text{sec}^2 x)\:dx in which both terms are trigonometric in nature. A useful idea is to consider which is the easier expression to integrate or differentiate. For our integral, we may decide to integrate \text{sec}^2 x and differentiate \text{tan} x.