Integrals and Application of Integrals
Integrals and Application of Integrals splits into technique (indefinite integrals) and evaluation (definite integrals and area). You need substitution, integration by parts using the ILATE order, partial fractions, and the standard forms for 1/(x² + a²), 1/√(x² ± a²) and 1/(a² − x²). Area questions use definite integrals between a curve and an axis, or between two curves.
Rules to remember
- ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫dx/x = log|x| + C.
- By parts: ∫u v dx = u∫v dx − ∫(u′∫v dx)dx, with u chosen by ILATE (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential).
- ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a − x)dx, the key symmetry property.
- If f is odd, ∫₋ₐᵃ f(x)dx = 0; if f is even, it equals 2∫₀ᵃ f(x)dx.
- Area under y = f(x) from a to b is ∫ₐᵇ |f(x)|dx; area between curves is ∫ₐᵇ (upper − lower)dx.
Worked example
Evaluate I = ∫₀^(π/2) sin x/(sin x + cos x)dx. Replacing x by 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. No antiderivative is needed; the property does the work.
Traps
- Omitting + C in indefinite integrals.
- Writing log x instead of log|x|.
- Forgetting to change limits, or to back-substitute, in a definite substitution.
- Reporting area below the x-axis as a negative number instead of taking the modulus.
- Subtracting the curves in the wrong order, which produces a negative area.