Integrals and Applications of Integrals
What is tested
This domain tests standard antiderivatives, substitution, integration by parts, partial fractions, definite-integral properties, and the computation of area under a curve or between two curves.
Core rules
- ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫dx/x = log|x| + C; ∫eˣ dx = eˣ + C.
- ∫dx/(a²+x²) = (1/a)tan⁻¹(x/a) + C; ∫dx/√(a²−x²) = sin⁻¹(x/a) + C.
- By parts: ∫uv dx = u∫v dx − ∫u′(∫v dx)dx, choosing u by the ILATE order (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential).
- Definite integrals: ∫₀^a f(x)dx = ∫₀^a f(a−x)dx; for odd f the integral over [−a,a] is 0, for even f it is 2∫₀^a f.
- Area = ∫_a^b |f(x)| dx; between curves, integrate upper minus lower.
Worked example
I = ∫₀^(π/2) sin x/(sin x + cos x) dx. Replace x by π/2 − x: 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.
Marks-losing traps
Dropping the constant C in indefinite integrals; forgetting to change limits (or substitute back) after u-substitution in a definite integral; missing the factor du = g′(x)dx; treating area below the x-axis as positive without the modulus; subtracting the curves in the wrong order.