Integrals
Indefinite and definite integration is the heaviest scoring domain. Questions cover standard forms, substitution, integration by parts, partial fractions, and definite-integral properties such as the King property. Expect one long 5–6 mark integral plus short evaluations.
Techniques
- Standard forms: ∫xⁿ dx = x^(n+1)/(n+1) + C, ∫dx/x = log|x| + C, ∫eˣ dx = eˣ + C, ∫dx/(1 + x²) = tan⁻¹x + C.
- Substitution: replace the inner function and adjust dx; for a definite integral also change the limits.
- By parts: ∫u v′ dx = uv − ∫u′v dx, choosing u by the ILATE order (inverse trig, log, algebraic, trig, exponential).
- Definite property: ∫₀^a f(x) dx = ∫₀^a f(a − x) dx.
- Fundamental theorem: ∫ₐ^b f(x) dx = F(b) − F(a).
Worked example
Let I = ∫₀^(π/2) sin x/(sin x + cos x) dx. Applying the property with x → π/2 − x gives I = ∫₀^(π/2) cos x/(cos x + sin x) dx. Adding the two forms, 2I = ∫₀^(π/2) 1 dx = π/2, so I = π/4.
Traps
- Dropping the constant of integration in indefinite integrals.
- Forgetting to change limits after substitution in a definite integral.
- Using log x rather than log|x| for ∫dx/x.
- Choosing the wrong u in integration by parts, which makes the integral harder.