What you'll learn
Integration as the reverse of differentiation and as accumulation/area.
Antiderivatives
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫1/x dx = ln|x| + C; ∫eˣ dx = eˣ + C; ∫cos x dx = sin x + C. Always include + C for indefinite integrals.
Riemann sums
Approximate area under a curve with rectangles (left, right, midpoint) or trapezoids. More subintervals → better approximation; the definite integral is the limit.
Fundamental Theorem of Calculus
- Part 1: d/dx ∫ₐˣ f(t) dt = f(x).
- Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is an antiderivative. The definite integral gives net (signed) area and total accumulated change.
u-substitution
Reverse the chain rule: let u = inner function, du = u′ dx, rewrite and integrate.
- Example: ∫2x·cos(x²) dx, u = x² → ∫cos u du = sin(x²) + C.
Exam tips
- Don't forget + C (indefinite) and to change limits (definite u-sub).
- Interpret a definite integral as accumulated change in context.
Common mistakes
- Omitting + C.
- Forgetting net area counts area below the axis as negative.