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Higher Mathematics

Indefinite integration

Reverse the power rule and include the constant of integration.

Topic practiceFormula supportMixed revision

Before you start

  • Differentiate powers confidently.
  • Understand that integration reverses differentiation.
  • Remember that a family of functions needs + C.
Higher Mathematics lesson

Explanation

Indefinite integration reverses the power rule. Increase the power by 1, divide by the new power and add C.

The exception n = −1 is usually handled separately, so avoid using the power rule there.

Visual support

tangentareaxy

Method and rules

  • ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1
  • ∫axⁿ dx = axⁿ⁺¹/(n + 1) + C
  • ∫k dx = kx + C

Worked examples

Worked example 1

Integrate a polynomial

Find ∫(6x² − 4x + 3) dx.

  1. Increase each power by 1.
  2. Divide by the new power.
  3. Integrate the constant as 3x.
  4. Add C.

Answer: 2x³ − 2x² + 3x + C

Worked example 2

Check by differentiating

Check ∫4x³ dx = x⁴ + C

  1. Differentiate x⁴ + C
  2. d/dx(x⁴) = 4x³ and d/dx(C) = 0

So: The derivative returns 4x³.

Worked example 3

Integrate a negative power

Find ∫6x⁻² dx.

  1. Increase −2 to −1.
  2. Divide by −1.

Answer: -6x⁻¹ + C

Watch out

  • Differentiating instead of integrating.
  • Forgetting + C.
  • Dividing by the old power instead of the new power.

Exam reminder

For indefinite integrals, + C is part of the answer. You only find a numerical C if a condition is supplied.