08 Integration by parts
The rule
- ∫ u dv = uv − ∫ v du, taken from the product rule d(uv) = u dv + v du.
- Choose u by LIATE: Logarithmic, Inverse trig, Algebraic (powers of x), Trigonometric, Exponential — the first type in this list becomes u; everything left, including dx, becomes dv.
- Use it when the integrand is a product of two unlike factors, e.g. x ex, x sin x, x ln x, or a single logarithm or inverse trig function.
Worked examples
- ∫ x ex dx: u = x, dv = exdx → x ex − ∫ ex dx = x ex − ex + C.
- ∫ x cos x dx: u = x, dv = cos x dx → x sin x − ∫ sin x dx = x sin x + cos x + C.
- ∫ ln x dx: u = ln x, dv = dx → x ln x − ∫ x·(1/x) dx = x ln x − x + C.
- ∫ x² ex dx needs parts twice: x²ex − 2x ex + 2ex + C.
- Choosing the wrong u (for instance u = ex in (1)) makes the new integral harder — always test whether ∫ v du is simpler than the original.
Choosing the method
1Sum of powers of x
Method
- Power rule, term by term
Example
- ∫ (3x² + 4x + 5) dx
2Composite function × derivative of the inner
Method
- Substitution
Example
- ∫ 2x(x² + 1)⁵ dx
3Derivative over the function itself
Method
- Logarithm
Example
- ∫ (2x + 1)/(x² + x + 4) dx
4Product of two unlike factors
Method
- Parts (LIATE)
Example
- ∫ x ln x dx
5sin²x or cos²x
Method
- Double-angle identity first
Example
- ∫ sin²x dx