Antiderivatives
Objective. Integrate powers and include constants of integration.
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Learn: the key idea
An antiderivative reverses differentiation, finding a function whose derivative is the given expression, and because derivatives of constants are zero, every antiderivative must include an arbitrary constant +c. The rule for powers is ∫xⁿ dx = xⁿ⁺¹/(n+1) + c, valid whenever n ≠ −1, raising the power by one and dividing by the new power. For example, ∫3x² dx = x³ + c, since differentiating x³ + c gives 3x² back. Each term of a sum is integrated separately, so ∫(4x + 5) dx = 2x² + 5x + c.
Worked example
Find the antiderivative of f(x) = 6x² − 4x + 1.
- 1Integrate 6x²: 6x³/3 = 2x³.
- 2Integrate −4x: −4x²/2 = −2x².
- 3Integrate 1: x.
- 4Combine terms and add the constant: F(x) = 2x³ − 2x² + x + c.
Answer: F(x) = 2x³ − 2x² + x + c
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Practice
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Start practisingCommon mistake. Students frequently forget the constant of integration, writing ∫2x dx = x² instead of x² + c. Without extra information fixing a point on the curve, infinitely many antiderivatives exist, differing only by this constant.
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