Integration by
Parts Calculator
Master the product rule in reverse. Solve complex integrals with ease.
Why Use This Calculator?
Integration by parts is one of the most powerful techniques in calculus. Our tool handles the heavy lifting.
Instant Results
Quickly find the antiderivative of products of functions.
Symbolic Logic
Uses advanced algorithms to apply the integration by parts formula correctly.
Learning Tool
Great for checking your manual work and understanding the technique.
What is Integration by Parts?
Integration by parts is a technique used to solve integrals where the integrand is a product of two functions. It is essentially the reverse of the product rule for differentiation.
The formula is derived from the product rule d(uv) = u dv + v du and is written as:
The goal is to choose u and dv such that the new integral ∫ v du is easier to solve than the original one.
The LIATE Rule
To choose the best function for u, follow the LIATE priority list. Choose the function that comes first in this list to be u:
- L Logarithmic functions (e.g., ln(x))
- I Inverse trigonometric functions (e.g., arctan(x))
- A Algebraic functions (e.g., x², 3x)
- T Trigonometric functions (e.g., sin(x))
- E Exponential functions (e.g., e^x)
How to Use This Calculator
- 1Enter the full function you want to integrate (e.g.,
x * e^x). - 2Select the variable of integration.
- 3Click "Calculate Solution". The calculator will automatically apply integration by parts if necessary.
Frequently Asked Questions
When should I use integration by parts?
Use it when the integrand is a product of two different types of functions, like x*sin(x) or x*e^x, and substitution doesn't work.
Can I use integration by parts for definite integrals?
Yes, the formula is [uv] from a to b minus the integral from a to b of v du.
What if integration by parts doesn't work?
Sometimes you need to apply it multiple times (e.g., for x^2 * e^x), or use other techniques like substitution or partial fractions.
Is this calculator free?
Yes, it is completely free to use.