Integration by Parts Formula
The integration by parts formula ∫ u dv = uv - ∫ v du is the reverse of the product rule.
The Formula
For definite integrals: .
When to use: The product rule for derivatives says . 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.
Quick Example
Notation
What This Formula Means
An integration technique based on the product rule: . Used when the integrand is a product of two functions.
The product rule for derivatives says . 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.
Formal View
Worked Examples
Example 1
easyAnswer
First step
Full solution
- 2 .
- 3 Factor: .
Example 2
hardExample 3
mediumCommon Mistakes
- Picking and backward so the new integral is harder — use LIATE so differentiates toward simpler.
- Dropping the minus sign or the term — the formula is , both pieces required.
- Trying parts when substitution fits — if an inner derivative is present, u-substitution is the right tool.
Why This Formula Matters
Integration by parts handles products that substitution can't — polynomial times exponential, anything times a logarithm or inverse trig. It encodes a strategic trade: you swap your integral for a new one, and choosing wisely (LIATE) makes the new integral easier, while a poor choice makes it harder. Recognizing it by "Is the integrand a product of unlike functions where differentiating one factor simplifies it, with no inner-derivative match for substitution?" — rather than by familiar numbers — is what lets a student tell it apart from u-substitution and product rule and liate choice of in a mixed problem set.
Frequently Asked Questions
What is the Integration by Parts formula?
An integration technique based on the product rule: . Used when the integrand is a product of two functions.
How do you use the Integration by Parts formula?
The product rule for derivatives says . 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.
What do the symbols mean in the Integration by Parts formula?
The LIATE rule helps choose : Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential—pick from earlier in this list.
Why is the Integration by Parts formula important in Math?
Integration by parts handles products that substitution can't — polynomial times exponential, anything times a logarithm or inverse trig. It encodes a strategic trade: you swap your integral for a new one, and choosing wisely (LIATE) makes the new integral easier, while a poor choice makes it harder. Recognizing it by "Is the integrand a product of unlike functions where differentiating one factor simplifies it, with no inner-derivative match for substitution?" — rather than by familiar numbers — is what lets a student tell it apart from u-substitution and product rule and liate choice of in a mixed problem set.
What do students get wrong about Integration by Parts?
The procedure for integration by parts is the easy part; the trap is picking and backward so the new integral is harder. Asking "Is the integrand a product of unlike functions where differentiating one factor simplifies it, with no inner-derivative match for substitution?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
What should I learn before the Integration by Parts formula?
Before studying the Integration by Parts formula, you should understand: integral, derivative.
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This formula is covered in depth in our complete guide:
How to Integrate Rational Functions: Long Division and Partial Fractions →