The indefinite integral
The indefinite integral ∫ f(x) dx is the family of all antiderivatives of f(x). It has no limits, so the answer is a function plus an arbitrary constant c.
Key rules used to integrate polynomials and simple bracket expressions:
- ∫ xn dx = xn+1/(n+1) + c, n ≠ −1
- ∫ a·f(x) dx = a ∫ f(x) dx
- ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
Key formula
For a linear bracket: ∫ (ax + b)n dx = (ax + b)n+1/[a(n+1)] + c, n ≠ −1.
Worked example
Worked example
Evaluate ∫ (2x + 3)4 dx. Here a = 2, n = 4, so the answer is (2x + 3)5/[2 × 5] + c = (2x + 3)5/10 + c.
Term-by-term integration
For a polynomial, integrate each term separately. Example: ∫ (6x² + 4x) dx = 6·x³/3 + 4·x²/2 + c = 2x³ + 2x² + c. Reversing, ∫ f(x) dx = x³ − 2x² + c means f(x) = 3x² − 4x, the derivative of the result.
Remember
- Always include + c.
- For (ax + b)n, divide by a as well as by the new power.