A-Level · A-Level Maths
A-Level Maths integration by parts
What is integration by parts?
Integration by parts is the integral analogue of the product rule: ∫ u dv/dx dx = uv − ∫ v du/dx dx. Use it when the integrand is a product where one factor simplifies under differentiation (polynomial × exponential, polynomial × trig, x × log x). The LIATE rule helps pick u: Logarithm, Inverse trig, Algebraic, Trig, Exponential — earlier in the list = better choice for u.
Worked example
Evaluate ∫ x · e^x dx.
- Apply LIATE: x is Algebraic, e^x is Exponential. Algebraic comes first, so let u = x and dv/dx = e^x.
- Differentiate u: du/dx = 1. Integrate dv/dx: v = e^x.
- Apply the formula: ∫ x · e^x dx = uv − ∫ v du/dx dx = x · e^x − ∫ e^x · 1 dx.
- Evaluate the remaining integral: x · e^x − e^x + C = e^x(x − 1) + C.
When to use integration by parts
Integration by parts is the right tool when:
- The integrand is a product of two different function types.
- One factor simplifies under differentiation (polynomials reduce in power; logarithms become rational).
- Substitution doesn't immediately work because there's no inner function to substitute.
The LIATE rule explained
LIATE is a heuristic for choosing u so that the resulting integral is simpler than the original. Logarithms first (because they simplify dramatically when differentiated), then Inverse trig, Algebraic (polynomials), Trig, Exponential. Pick u from earliest in the list.
Where it appears in the exam
AQA 7357, Edexcel 9MA0 and OCR H240 all examine integration by parts in Pure papers. Definite-integral questions where the integrand is x times a trig or exponential function are the most common form.
Common mistakes
- Choosing the wrong u — the resulting integral becomes harder, not easier.
- Sign errors when applying the formula: ∫ u dv/dx dx = uv − ∫ v du/dx dx (the minus is non-negotiable).
- Forgetting +C on indefinite integrals — easy 1-mark loss.
- Stopping after one application when the integral needs IBP applied twice (e.g., x² · e^x).
Frequently asked
- Can I always use integration by parts?
- No — IBP works when the integrand is a product of two function types where one simplifies under differentiation. For nested composites, substitution is usually better.
- Does the order of u and dv/dx matter?
- Yes. The wrong choice produces a remaining integral that's harder than the original. The LIATE rule is the fastest heuristic for picking u.
- How do I integrate ln(x) using by parts?
- Treat ln(x) as ln(x) × 1. Let u = ln(x) and dv/dx = 1. Then du/dx = 1/x and v = x. Apply the formula: ∫ ln(x) dx = x · ln(x) − ∫ x · (1/x) dx = x · ln(x) − x + C.
A-Level Maths glossary terms
- Integration by partsIntegration by parts is the integral analogue of the product rule: ∫ u dv/dx dx = uv − ∫ v du/dx dx. Use it whenever the integrand is a product of two functions where one becomes simpler under differentiation (typically polynomial × exponential, polynomial × trig, or x × log x). The LIATE rule helps pick u: Logarithm, Inverse trig, Algebraic, Trig, Exponential.
- Chain ruleThe chain rule differentiates composite functions: if y = f(g(x)) then dy/dx = f'(g(x)) · g'(x). It appears across A-Level Maths and Further Maths whenever an outer function wraps an inner function — common examples are sin(2x), e^(3x²) and (1+x²)⁵. Without the chain rule, every composite differentiation drops marks; with it, the standard trick is to spot the inner function first.
- Differential equationsA differential equation relates a function to its derivatives. A-Level Maths examines first-order separable equations (dy/dx = f(x)·g(y)), solved by separating variables and integrating both sides. Applications include exponential growth/decay (dN/dt = kN), Newton's law of cooling and rate-of-reaction models. Further Maths extends to second-order equations and integrating factors.
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