Integral Calculus
Integral calculus in the IPE papers covers indefinite integration by standard forms, substitution and integration by parts, partial fractions, trigonometric integrals and reduction formulae, plus definite integrals and their symmetry properties, ending with area under a curve.
Core rules and concepts
- Standard forms: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫dx/x = ln|x| + C; ∫eˣ dx = eˣ + C.
- By parts: ∫u dv = uv − ∫v du, with ILATE fixing the order of preference for u.
- Definite integral properties: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx; ∫₋ₐᵃ f = 0 for odd f and 2∫₀ᵃ f for even f.
- Substitution in a definite integral requires changing the limits at the same time.
- Area between a curve and the x-axis is ∫|y| dx over the stated interval.
Worked example
Evaluate ∫₀^π x sin x dx. Take u = x, dv = sin x dx, so du = dx, v = −cos x. Then the integral equals [−x cos x]₀^π + ∫₀^π cos x dx. First term: (−π cos π) − 0 = π. Second term: [sin x]₀^π = 0. Value = π.
Traps that cost marks
- Omitting the constant of integration in an indefinite integral.
- Keeping the old limits after a substitution.
- Sign slips when v = −cos x is multiplied by u.
- Writing ∫dx/(1 + x²) with a wrong standard result; it is tan⁻¹x + C.
- Splitting an integral across a point where the integrand is undefined without treating it as improper.