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.
Which is u and which is dv: the LIATE rule
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.
dv is everything that is left over, including the dx, and it has to be something you can integrate. If a single function is all there is, like ln x on its own, then u is that function and dv is just dx.
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.
Two more examples of choosing u and dv
The integral of x cos x dx: x is algebraic and cos x is trig, so u = x and dv = cos x dx. Then du = dx and v = sin x, which gives x sin x minus the integral of sin x dx, so the answer is x sin x + cos x + C.
The integral of ln x dx: there is only one function, and logarithms come first in LIATE, so u = ln x and dv = dx. Then du = (1/x) dx and v = x, which gives x ln x minus the integral of 1 dx, so the answer is x ln x - x + C.
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.
Which is u and which is dv in integration by parts?
u is the factor that comes first in LIATE. dv is everything else, including dx. In the integral of x e^x dx, u = x and dv = e^x dx. In the integral of ln x dx, u = ln x and dv = dx.