Integration as the reverse of differentiation
Integration reverses differentiation. An indefinite integral includes an arbitrary constant of integration C, because differentiating a constant gives zero.
Key idea
Power rule: ∫ xn dx = xn+1⁄(n + 1) + C, for n ≠ −1. For a linear bracket, ∫ (ax + b)n dx = (ax + b)n+1⁄[a(n + 1)] + C.
Standard integrals and definite integrals
Useful results are ∫ ex dx = ex + C, ∫ (1⁄x) dx = ln|x| + C, ∫ cos x dx = sin x + C and ∫ sin x dx = −cos x + C. A definite integral uses limits and gives a number: no constant is needed.
Worked example
Evaluate ∫ from 1 to 3 of 2x dx. The antiderivative is x2, so the value is [x2] from 1 to 3 = 9 − 1 = 8.
Remember
- Add 1 to the power and divide by the new power.
- Always include + C for indefinite integrals.
- For (ax + b)n divide by the extra factor a.