Integration and Definite Integrals
Integration questions test standard antiderivatives, simple substitution and evaluation of definite integrals using limits and their properties. The marks come from correct constants, correct limits and clean arithmetic rather than from long manipulation.
Indefinite integrals
Integration reverses differentiation, so every indefinite integral needs +c. Standard results: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1; ∫(1/x)dx = log|x| + c; ∫eˣ dx = eˣ + c; ∫aˣ dx = aˣ/(log a) + c; ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n+1)] + c. Constants factor out and sums split.
Substitution
Substitute u for the inner function or denominator when its derivative (or a constant multiple) appears in the integrand. Choosing u is the whole skill; once chosen, convert dx as well.
Definite integrals
∫ₐᵇ f(x)dx = F(b) − F(a), where F is any antiderivative, so no +c is needed. Properties: ∫ₐᵇ = −∫ᵦᵃ; ∫ₐᵇ = ∫ₐᶜ + ∫꜀ᵇ; ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a − x)dx.
Worked example
Evaluate ∫₁² (2x + 3)/(x² + 3x + 5) dx. The numerator is exactly the derivative of the denominator, so the antiderivative is log(x² + 3x + 5). At x = 2 the denominator is 15; at x = 1 it is 9. Value = log 15 − log 9 = log(15/9) = log(5/3). The quadratic is positive on [1, 2], so the modulus is unnecessary.
Traps
- Omitting +c in an indefinite integral.
- Writing ∫(1/x)dx = log x and ignoring the modulus.
- Forgetting to change the limits when substituting in a definite integral.
- Applying the power rule to ∫(1/x)dx, which is the one excluded case.