Integration is the reverse of differentiation; it finds the accumulated total or area under a curve, essential for cost, revenue, and supply–demand problems.
## Core concept
Integration (also called anti-differentiation) reverses differentiation. If `d/dx[f(x)] = g(x)`, then `∫g(x)dx = f(x) + C`, where C is the constant of integration.
In business: - Total cost from marginal cost: integrate MC to find TC - Total revenue from marginal revenue: integrate MR to find TR - Area under demand curve: consumer surplus - Accumulated growth: population, investment returns over time
The constant C appears because differentiation of a constant is zero—multiple functions share the same derivative.
Definite integrals `∫ₐᵇ f(x)dx` compute the net accumulated value between limits a and b (often time or quantity bounds).
## Formula / rule
| Rule | Formula | |------|---------| | Power rule | `∫xⁿ dx = xⁿ⁺¹/(n+1) + C` (n ≠ −1) | | Constant multiple | `∫kf(x)dx = k∫f(x)dx` | | Sum/difference | `∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx` | | Definite integral | `∫ₐᵇ f(x)dx = [F(b)] − [F(a)]` where F is antiderivative | | Exponential | `∫eˣ dx = eˣ + C` | | Logarithm | `∫(1/x)dx = ln\|x\| + C` |
Definite integral evaluation: Find the antiderivative F(x), then substitute the upper limit and lower limit, and subtract.
## Common exam applications
1. Total cost from marginal cost - Given: MC = 5 + 3x (where x = units) - TC = ∫MC dx = ∫(5 + 3x)dx = 5x + 3x²/2 + C - If fixed cost (FC) = ₹200, then C = 200, so TC = 5x + 1.5x² + 200
2. Consumer surplus - Demand curve: p = 100 − 2x - At equilibrium x = 10, p = 80 - CS = ∫₀¹⁰(100 − 2x)dx − (80 × 10) - = [100x − x²]₀¹⁰ − 800 - = 1000 − 100 − 800 = ₹100
3. Accumulated revenue over time - Revenue flow: R(t) = 1000 + 50t per month - Total over 12 months: ∫₀¹²(1000 + 50t)dt = [1000t + 25t²]₀¹² - = 12000 + 25(144) = ₹15,600
## Common mistakes
- Forgetting the constant of integration (C) in indefinite integrals—this is only omitted when solving definite integrals.
2. Wrong application of power rule: ∫x⁻¹ dx ≠ x⁰/0. Correct: ∫(1/x)dx = ln|x| + C.
3. Sign errors in definite integrals: Always subtract F(lower) from F(upper), not reversed.
4. Misidentifying limits: In a business problem, ensure limits match the variable (e.g., if x is units, limits are quantities; if t is time, limits are time periods).
5. Ignoring boundary conditions: When finding C, use given initial values (e.g., TC at x = 0 equals fixed cost).
6. Treating integration like differentiation: ∫(f + g) ≠ ∫f + ∫g is false. It is true—splitting is allowed.
## Quick checklist for exam
- [ ] Identify if you need indefinite (+ C) or definite (a to b) integral
- [ ] Check if the integrand matches a standard form (power, exponential, logarithm)
- [ ] Apply linearity: split sums and factor out constants
- [ ] Use antiderivative formula correctly
- [ ] For definite integrals, substitute upper limit first, then lower, then subtract
- [ ] Check units and context (cost in ₹, revenue in ₹, etc.)