Q1. Evaluate ∫(3x² + 2x) dx.
Apply the power rule to each term
∫3x² dx = x³, ∫2x dx = x²
Result = x³ + x² + C
Integrals — Class 12
Integration reverses differentiation. Learn the standard formulas and the fundamental theorem that turns them into definite areas.
Indefinite integral∫f(x)dx = F(x) + C, where F'(x) = f(x) and C is the arbitrary constant of integration.
Definite integral∫ₐᵇ f(x)dx = F(b) − F(a) — the fundamental theorem of calculus, giving a number (often an area).
Integration by substitutionReplace a part of the integrand with a new variable u to simplify the integral.
Solved examples
Q2. Evaluate ∫₀² x dx.
∫x dx = x²/2
Apply limits: [x²/2] from 0 to 2
= 2²/2 − 0²/2 = 2
Formula sheet
| Quantity | Formula |
|---|---|
| Power rule | ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠−1) |
| Exponential | ∫eˣ dx = eˣ + C |
| Reciprocal | ∫(1/x) dx = ln|x| + C |
| Fundamental theorem | ∫ₐᵇ f(x)dx = F(b) − F(a) |
Common mistakes & exam wins
- Never forget the "+C" for an indefinite integral — it represents the whole family of antiderivatives.
- A definite integral has NO constant of integration — the C cancels out when you subtract F(a) from F(b).
- Check your answer by differentiating it — you should get back the original integrand.
Frequently asked questions
What is the difference between a definite and indefinite integral?
An indefinite integral gives a family of functions (with a +C); a definite integral gives a single number, F(b) − F(a).
What is the power rule for integration?
∫xⁿ dx = xⁿ⁺¹/(n+1) + C, valid for all n ≠ −1.
What is the fundamental theorem of calculus?
It connects differentiation and integration: ∫ₐᵇ f(x)dx = F(b) − F(a), where F is any antiderivative of f.