Integration is the reverse of differentiation; it finds the antiderivative (or primitive) of a function and is used to calculate areas under curves, accumulation, and problems involving rates of change.
## Core concept
Integration answers: "What function, when differentiated, gives me this expression?"
- Indefinite integral: ∫f(x)dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration
- Constant of integration C: Always include it in indefinite integrals—it accounts for all possible vertical shifts of the antiderivative
- Integrand: The function being integrated (the expression after ∫)
- Integration reverses differentiation: If d/dx[x³] = 3x², then ∫3x² dx = x³ + C
## Formula / rule
Power rule for integration: - ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ −1 - ∫x⁻¹ dx = ln|x| + C (special case; reciprocal of x)
Basic standard integrals (CA Foundation scope): - ∫k dx = kx + C (k is constant) - ∫eˣ dx = eˣ + C - ∫aˣ dx = (aˣ)/(ln a) + C, where a > 0, a ≠ 1 - ∫(1/x) dx = ln|x| + C - ∫sin x dx = −cos x + C - ∫cos x dx = sin x + C - ∫sec² x dx = tan x + C
Linearity of integration: - ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx - ∫k·f(x) dx = k∫f(x) dx (k is constant)
## Common exam applications
1. Finding an antiderivative: Given a marginal cost or revenue function, integrate to find total cost or revenue.
2. Initial condition problems: Use given conditions to find the value of C.
3. Area under a curve: Set up for definite integrals (foundation for future work).
4. Accumulation in economics: Total profit = ∫(marginal profit) dx
## Worked example
Q: A firm's marginal cost function is MC = 5x² + 3x + 2. Find the total cost function if fixed cost = ₹100.
Solution: - TC = ∫MC dx = ∫(5x² + 3x + 2) dx - TC = 5·(x³/3) + 3·(x²/2) + 2x + C - TC = (5x³)/3 + (3x²)/2 + 2x + C - At x = 0, TC = fixed cost = 100, so C = 100 - TC = (5x³)/3 + (3x²)/2 + 2x + 100
## Common mistakes
- Forgetting the constant C: Every indefinite integral must include + C. Without it, marks are lost in exam.
- Wrong power rule application: ∫x³ dx ≠ 3x⁴ + C. Correct: ∫x³ dx = x⁴/4 + C
- Confusing ∫(1/x) with ∫x⁻¹: Both are ln|x| + C, not ln x without absolute value.
- Not simplifying the integrand first: Before integrating, expand brackets and split fractions. ∫[(x+1)²/x] dx should be expanded first.
- Misplacing the +C: Write +C *outside* any evaluation—it cancels in definite integrals but is essential in indefinite ones.
Key exam tips: - Differentiate your answer to verify: d/dx of your result should equal the integrand. - Practice standard integrals until they are automatic. - In word problems, read carefully to identify whether you need indefinite (general) or definite (specific) integration.