Integration by Parts, Made Intuitive
Updated 2026-07-12
Integration by parts is a technique for integrating a product of two functions. It comes from the product rule in reverse, and its formula is: the integral of u dv equals u v minus the integral of v du. You pick one factor as u and the rest as dv.
Integration by parts is the tool for integrals that are a product of two different kinds of function, like x times e^x. The whole game is choosing which factor to call u, and there is a simple rule of thumb for that.
The formula and where it comes from
The formula says the integral of u dv equals u v minus the integral of v du. It is just the product rule for derivatives, integrated and rearranged, so it trades one integral for another that is meant to be easier.
Choosing u with LIATE
Pick u using the order Logarithmic, Inverse trig, Algebraic, Trig, Exponential. Whichever type appears first in that list becomes u, and the rest becomes dv.
The goal is to pick u so that its derivative gets simpler. Algebraic terms like x become simpler when differentiated, which is why they are usually u.
Worked example
Evaluate the integral of x e^x dx.
- By LIATE, the algebraic factor x comes before the exponential, so u = x and dv = e^x dx.
- Then du = dx and v = e^x.
- Apply the formula: u v minus the integral of v du, which is x e^x minus the integral of e^x dx.
Answer: x e^x - e^x + C.
The common mistake
Wrong: Choosing u and dv so the new integral is harder, not easier.
Fix: Use LIATE. If your remaining integral looks worse than the original, you likely swapped u and dv.
Wrong: Forgetting the minus sign in front of the second integral.
Fix: The formula is u v minus the integral of v du. The minus is not optional, and dropping it is the most common arithmetic error.
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Frequently asked questions
What is the integration by parts formula?
The integral of u dv equals u v minus the integral of v du. You split the integrand into a part called u and a part called dv, then apply the formula.
How do you choose u in integration by parts?
Use LIATE: Logarithmic, Inverse trig, Algebraic, Trig, Exponential. Whichever appears first becomes u, so that differentiating u makes the problem simpler.