Integration by parts is useful when an integral contains a product of functions and direct integration is inconvenient. The Integration by Parts Calculator applies the relationship that connects the original product with a new, often simpler, integral.
Formula
The standard formula is ∫u dv = uv − ∫v du. The main decision is choosing u and dv so that differentiating u makes it simpler and integrating dv is manageable.
Choosing u and dv
For a product such as x eˣ, a common choice is u=x and dv=eˣdx. Then du=dx and v=eˣ, so the integral becomes xeˣ−∫eˣdx = eˣ(x−1)+C.
How to Use the Calculator
Enter the integral and, if the calculator asks for them, specify the u and dv portions. Review the returned intermediate terms because an algebraically correct integration-by-parts setup can still become unnecessarily complicated if the choices are poor.
Definite Integrals
For definite integrals, the same identity is applied with limits: ∫ₐᵇu dv = [uv]ₐᵇ − ∫ₐᵇv du. The endpoint evaluation is part of the calculation, so the constant of integration is not included.
Best Way to Check
Differentiate an indefinite result or evaluate the endpoint expression again for a definite integral. This catches sign errors, especially the minus sign before the second integral.