The antiderivative and the indefinite integral

Definition

• An antiderivative (inverse derivative) of f(x) on an interval is a function F(x) whose derivative is f(x), that is F′(x) = f(x) for every x in the interval.
• Adding any constant to an antiderivative gives another antiderivative, so every function has infinitely many antiderivatives.
• The set of all antiderivatives is called the indefinite integral and is written ∫ f(x) dx: ∫ f(x) dx = F(x) + C.
• Geometrically the indefinite integral is a family of parallel curves differing only by the constant, so at the same value of x their tangents all have the same slope.
• C is the constant of integration (arbitrary constant); it is fixed when a point on the curve is known.

Properties

• Integral of a sum or difference: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.
• Constant factor: ∫ k f(x) dx = k ∫ f(x) dx, where k is a fixed real number.
• Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1. The excluded case is exactly the logarithmic one: ∫ dx/x = ln|x| + C.
• Check any result by differentiating: d/dx of the answer must return the integrand.

Worked examples

1. ∫ (3x² + 4x + 5) dx = x³ + 2x² + 5x + C.
2. ∫ (x − 1)(x + 3) dx → expand first: ∫ (x² + 2x − 3) dx = x³/3 + x² − 3x + C.
3. ∫ (√x + 1/√x) dx = ∫ (x^1/2 + x^−1/2) dx = (2/3)x^3/2 + 2x^1/2 + C.
4. ∫ x(x² − 3)² dx: substitution is the efficient route (section 07) — u = x² − 3, du = 2x dx → ½∫ u² du = (x² − 3)³/6 + C. Expanding first also works: ∫ (x⁵ − 6x³ + 9x) dx = x⁶/6 − 3x⁴/2 + 9x²/2 + C, which differs from the factored form only by a constant.