Integration by Parts
Integration by parts formula: the integral of u dv equals uv minus the integral of v du.
Essential for integrating products of functions.
The Formula
Integration by parts is the reverse of the product rule for differentiation. It transforms a difficult integral into a simpler one by strategically choosing which part of the integrand to differentiate and which to integrate.
The key to success is choosing u and dv wisely. The LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) helps you pick u — choose the function type that comes first in this list.
Variables
| Symbol | Meaning |
|---|---|
| u | The function you choose to differentiate |
| dv | The remaining part of the integrand (which you integrate) |
| du | The derivative of u |
| v | The antiderivative of dv |
The LIATE Rule
| Priority | Function Type | Example |
|---|---|---|
| 1 (choose first as u) | Logarithmic | ln(x), log(x) |
| 2 | Inverse trigonometric | arctan(x), arcsin(x) |
| 3 | Algebraic | x², x, 3x+1 |
| 4 | Trigonometric | sin(x), cos(x) |
| 5 (choose last as u) | Exponential | eˣ, 2ˣ |
Example 1
Evaluate ∫ x eˣ dx
Choose u = x (algebraic) and dv = eˣ dx (exponential)
Then du = dx and v = eˣ
Apply the formula: ∫ x eˣ dx = xeˣ − ∫ eˣ dx
= xeˣ − eˣ + C
∫ x eˣ dx = eˣ(x − 1) + C
Example 2
Evaluate ∫ x² ln(x) dx
Choose u = ln(x) (logarithmic, higher LIATE priority) and dv = x² dx
Then du = (1/x) dx and v = x³/3
Apply the formula: ∫ x² ln(x) dx = (x³/3) ln(x) − ∫ (x³/3)(1/x) dx
= (x³/3) ln(x) − ∫ x²/3 dx
= (x³/3) ln(x) − x³/9 + C
∫ x² ln(x) dx = (x³/3)(ln(x) − 1/3) + C
When to Use It
Use integration by parts when the integrand is a product of two different types of functions.
- Products of polynomials and exponentials (like x²eˣ)
- Products of polynomials and trigonometric functions (like x sin(x))
- Products involving logarithms (like x ln(x))
- Inverse trigonometric functions (like arctan(x))
- Sometimes applied twice in a row (tabular integration for repeated polynomial factors)
Tip: if applying integration by parts once gives you an integral that is harder than the original, try swapping your choice of u and dv.