Integration by Parts

Calculus
process

Also known as: IBP, parts method, integration-polar

Grade 9-12

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An integration technique based on the product rule: \int u\,dv = uv - \int v\,du. Essential for integrating products like x\sin x, x^2 e^x, \ln x, and e^x\cos x.

This concept is covered in depth in our Integration of Rational Functions Guide, with worked examples, practice problems, and common mistakes.

Definition

An integration technique based on the product rule: \int u\,dv = uv - \int v\,du. Used when the integrand is a product of two functions.

💡 Intuition

The product rule for derivatives says (uv)' = u'v + uv'. Rearranging and integrating gives integration by parts. The idea is to trade your original integral for a (hopefully easier) one. You're transferring the derivative from one factor to the other.

🎯 Core Idea

Integration by parts converts \int u\,dv into uv - \int v\,du. The goal is to choose u and dv so that \int v\,du is simpler than the original integral.

Example

\int x\,e^x\,dx Let u = x, dv = e^x\,dx. Then du = dx, v = e^x.
= xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x - 1) + C

Formula

\int u\,dv = uv - \int v\,du
For definite integrals: \int_a^b u\,dv = [uv]_a^b - \int_a^b v\,du.

Notation

The LIATE rule helps choose u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential—pick u from earlier in this list.

🌟 Why It Matters

Essential for integrating products like x\sin x, x^2 e^x, \ln x, and e^x\cos x. Also used to derive reduction formulas and in proving theoretical results. Together with u-substitution, it handles the majority of integration problems.

💭 Hint When Stuck

Use LIATE to pick u, then set up a two-column table: u and dv on top, du and v on the bottom.

Formal View

If u and v are differentiable on [a, b], then \int_a^b u(x) v'(x)\,dx = [u(x) v(x)]_a^b - \int_a^b u'(x) v(x)\,dx. Indefinite form: \int u\,dv = uv - \int v\,du.

Related Concepts

IntegralDerivative

🚧 Common Stuck Point

Sometimes you need to apply integration by parts TWICE (as with \int e^x\sin x\,dx) and then solve for the original integral algebraically when it reappears.

⚠️ Common Mistakes

  • Choosing u and dv poorly: if you let u = e^x and dv = x\,dx in \int xe^x\,dx, the new integral \int \frac{x^2}{2}e^x\,dx is HARDER. Use LIATE: pick u = x (algebraic) and dv = e^x\,dx.
  • Forgetting the minus sign: the formula is uv \mathbf{-} \int v\,du. Dropping the negative is a common algebraic error that changes the entire answer.
  • Not recognizing when to stop: for \int e^x\cos x\,dx, after applying IBP twice, the original integral reappears—set up an equation and solve for it instead of continuing to integrate by parts forever.

Frequently Asked Questions

What is Integration by Parts in Math?

An integration technique based on the product rule: \int u\,dv = uv - \int v\,du. Used when the integrand is a product of two functions.

Why is Integration by Parts important?

Essential for integrating products like x\sin x, x^2 e^x, \ln x, and e^x\cos x. Also used to derive reduction formulas and in proving theoretical results. Together with u-substitution, it handles the majority of integration problems.

What do students usually get wrong about Integration by Parts?

Sometimes you need to apply integration by parts TWICE (as with \int e^x\sin x\,dx) and then solve for the original integral algebraically when it reappears.

What should I learn before Integration by Parts?

Before studying Integration by Parts, you should understand: integral, derivative.

Prerequisites

integralderivative

How Integration by Parts Connects to Other Ideas

To understand integration by parts, you should first be comfortable with integral and derivative.

Want the Full Guide?

This concept is explained step by step in our complete guide:

How to Integrate Rational Functions: Long Division and Partial Fractions →
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