Integration by parts — LIATE rule

Martin McBride
Technological Singularity
9 min readMay 20, 2024

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The technique of integration by parts can often be used for solving complex integrals, particularly if we need to integrate a function that can be expressed as the product of two other functions.

The basic integration by parts formula is:

However, it can sometimes be difficult to decide how best to proceed.

The LIATE rule is a simple rule of thumb to help with this. It isn’t foolproof, so we should always be prepared to try a different route if the LIATE rule doesn’t give a good result. Indeed, there is not even consensus on the order of the rule, and some people use the ILATE rule (the slight difference will become clear soon).

Also, of course, the LIATE rule only covers certain classes of functions. These include all the very common mathematical functions, but if we ever need to integrate an expression involving other types of functions, the rule won’t directly help us. But we can still apply the principles of these rules to other function types.

In this article, we will explain the rule and give an example of each of the cases.

The LIATE rule

The LIATE rule recognises five main types of function that commonly occur in integration problems:

  • Logarithmic functions (L), eg ln(x).
  • Inverse trig functions (I), eg arcsin(x).
  • Algebraic functions (A), eg .
  • Trigonometric functions (T), eg sin(x).

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