Integration by Parts
Integration by parts is an integration technique derived from the product rule of differentiation, used to integrate products of two functions. The formula transforms an integral of a product into a simpler expression by strategically choosing which factor to differentiate (u) and which to integrate (dv). It is particularly effective for integrands involving products of polynomials with trigonometric, exponential, or logarithmic functions.
Key Formula
∫ u dv = uv − ∫ v du
LaTeX: \int u\, dv = uv - \int v\, du
| Symbol | Meaning | Unit |
|---|---|---|
| u | First function (to differentiate) | dimensionless |
| dv | Differential of second function (to integrate) | dimensionless |
| v | Antiderivative of dv | dimensionless |
| du | Derivative of u times dx | dimensionless |
Worked Example
Problem
Evaluate ∫ x · eˣ dx.
Solution
Step 1: Choose u and dv using the LIATE rule (Logarithm, Inverse trig, Algebraic, Trig, Exponential — choose u from earlier in list). Let u = x → du = dx. Let dv = eˣ dx → v = eˣ. Step 2: Apply the formula: ∫ u dv = uv − ∫ v du. ∫ x·eˣ dx = x·eˣ − ∫ eˣ dx. Step 3: Evaluate remaining integral: ∫ eˣ dx = eˣ. Step 4: Combine: x·eˣ − eˣ + C = eˣ(x − 1) + C.
Answer
∫ x·eˣ dx = eˣ(x − 1) + C
LIATE Rule for Choosing u in Integration by Parts
| Priority | Function Type | Symbol | Example u | Reason |
|---|---|---|---|---|
| 1st (highest) | Logarithmic | L | ln(x), log(x) | Hard to integrate alone |
| 2nd | Inverse Trigonometric | I | arcsin(x), arctan(x) | Hard to integrate alone |
| 3rd | Algebraic | A | x², x³, polynomials | Simplifies when differentiated |
| 4th | Trigonometric | T | sin(x), cos(x) | Easy to integrate |
| 5th (lowest) | Exponential | E | eˣ, 2ˣ | Unchanged by integration |
Interactive Tools
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Related Terms
Integration by Substitution
Integration by substitution is a technique that simplifies complex integrals by replacing a part of the integrand with a new variable, effectively reversing the chain rule of differentiation. The method works by identifying an inner function u = g(x), computing du = g'(x) dx, and rewriting the integral entirely in terms of u. It is one of the most widely used integration techniques and is applicable whenever an integrand contains a composite function paired with its derivative.
Integral
An integral is a mathematical object that represents the accumulation of quantities, such as areas under curves, total displacement, or accumulated change. There are two main types: the definite integral, which yields a numerical value representing the net area between a function and the x-axis over an interval, and the indefinite integral, which yields a family of antiderivative functions. Integration is the reverse process of differentiation and is one of the two fundamental operations of calculus.
The phrase "integration by parts" reflects the splitting of an integral into two "parts" using the product rule. The method was formalized by Brook Taylor in 1715 and stems directly from Leibniz's product rule for differentiation developed in the 1670s.