Maths
Integration (calculus) — quick study summary
Integration is the reverse of differentiation: given a function's derivative, find the original function (the 'antiderivative'). Indefinite integrals have a constant + C; definite integrals evaluate the area under a curve between two limits. Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1). The fundamental theorem connects differentiation and integration. Common techniques: substitution (reverse chain rule), integration by parts (reverse product rule).
Key points
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
- ∫1/x dx = ln|x| + C
- ∫eˣ dx = eˣ + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C
- Definite integral ∫ᵃᵇ f(x) dx = F(b) − F(a) (signed area under curve)
- Integration by parts: ∫u dv = uv − ∫v du
Practice quiz
Click each question to reveal the answer.
1. What is ∫(3x² + 2x) dx?
- x³ + x² + C
- 6x + 2 + C
- x³ + x²
- 3x³ + 2x² + C
Answer: x³ + x² + C
Apply the power rule term by term: ∫3x² = x³, ∫2x = x², plus constant C.
2. Evaluate ∫₀² 2x dx
Answer: 4
Antiderivative is x². Evaluate: 2² − 0² = 4.
3. What does the constant of integration C represent geometrically?
Answer: A vertical shift — there are infinitely many antiderivatives differing by a constant
All antiderivatives of f(x) form a family of parallel curves shifted up or down — they all have the same derivative.
Last reviewed: May 2026