The LIPET Strategy for Integration by Parts

Integration by parts is one of many integration techniques that are used in calculus. This method of integration can be thought of as a way to undo the product rule. One of the difficulties in using this method is determining what function in our integrand should be matched to which part. The LIPET acronym can be used to provide some guidance on how to split up the parts of our integral.

Integration by Parts

Recall the method of integration by parts. The formula for this method is:

u dv = uv - ∫ v du.

This formula shows which part of the integrand to set equal to u, and which part to set equal to dv. LIPET is a tool that can help us in this endeavor.

The LIPET Acronym

The word “LIPET” is an acronym, meaning that each letter stands for a word. In this case, the letters represent different types of functions. These identifications are:

  • L = Logarithmic function
  • I = Inverse trigonometric function
  • P = Polynomial function
  • E = Exponential function
  • T = Trigonometric function

This gives a systematic list of what to try to set equal to u in the integration by parts formula. If there is a logarithmic function, try setting this equal to u, with the rest of the integrand equal to dv. If there are no logarithmic or inverse trig functions, try setting a polynomial equal to u. The examples below help to clarify the use of this acronym.

Example 1

Consider ∫ x lnx dx. Since there is a logarithmic function, set this function equal to u = ln x. The rest of the integrand is dv = x dx. It follows that du = dx / x and that v = x2/ 2.

This conclusion could be found by trial and error. The other option would have been to set u = x. Thus du would be very easy to calculate. The problem arises when we look at dv = lnx. Integrate this function in order to determine v. Unfortunately, this is a very difficult integral to calculate.

Example 2

Consider the integral ∫ x cos x dx. Start with the first two letters in LIPET. There are no logarithmic functions or inverse trigonometric functions. The next letter in LIPET, a P, stands for polynomials. Since the function x is a polynomial, set u = x and dv = cos x.

This is the correct choice to make for integration by parts as du = dx and v = sin x. The integral becomes:

x sin x - ∫ sin x dx.

Obtain the integral through a straightforward integration of sin x.

When LIPET Fails

There are some cases where LIPET fails, which requires setting u equal to a function other than the one prescribed by LIPET. For this reason, this acronym should only be thought of as a way to organize thoughts. The acronym LIPET also provides us with an outline of a strategy to try when using integration by parts. It is not a mathematical theorem or principle that is always the way to work through an integration by parts problem.