Integration
Core ideas
Indefinite and definite integrals, standard forms, substitution, integration by parts, partial fractions, and trigonometric identities. Applications include areas under curves, areas between curves, volumes of revolution, and solving simple differential equations by separation of variables. The fundamental theorem of calculus links definite integrals to antiderivatives.
Traps
- Omitting the constant of integration in indefinite integrals.
- With substitution, change the limits for definite integrals, or convert back to x before evaluating.
- Integration by parts: choose u so that du simplifies; LIATE (log, inverse trig, algebraic, trig, exponential) is a useful ordering.
- Areas below the x-axis give negative integrals; take the modulus or split the region.
- Volumes of revolution use pi times the integral of y^2, not y.
Rule of thumb
For substitution, let u equal the inner function whose derivative appears (up to a constant) elsewhere in the integrand. Then du = u' dx and the integral transforms cleanly.
Worked example
Evaluate the integral from 0 to 1 of x e^(x^2) dx. Let u = x^2, du = 2x dx, so x dx = du/2. Limits: x=0 gives u=0; x=1 gives u=1. The integral becomes (1/2) times the integral from 0 to 1 of e^u du = (1/2)[e^u] from 0 to 1 = (e - 1)/2.