Integrals
Integrals
The exam tests indefinite and definite integration using standard forms, substitution, integration by parts, partial fractions and properties of definite integrals.
Core rules
- Standard forms: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C; ∫eˣ dx = eˣ + C.
- Substitution reverses the chain rule; change dx with du and finish in the original variable for indefinite integrals.
- By parts: ∫u dv = uv − ∫v du. Choose u using ILATE (inverse trig, log, algebraic, trig, exponential).
- Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a). When substituting, change the limits instead of returning to x.
- Useful properties: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx; for odd f over [−a,a], the integral is zero, and for even f it is 2∫₀ᵃ f.
Worked example
Evaluate ∫x eˣ dx.
Use integration by parts with u = x and dv = eˣ dx. Then du = dx and v = eˣ. So ∫x eˣ dx = x eˣ − ∫eˣ dx = x eˣ − eˣ + C = eˣ(x − 1) + C. Differentiating eˣ(x − 1) gives eˣ(x − 1) + eˣ = x eˣ, which checks the answer.
Traps
- Omitting +C in an indefinite integral.
- Losing a minus sign in the by-parts formula, especially with trigonometric factors.
- Forgetting to change limits after substitution in a definite integral.
- Writing ∫(1/x) dx = ln x instead of ln|x|.
- Applying an even/odd shortcut without checking that the interval is symmetric.