Higher Maths worked question

Integrate a polynomial and include the constant

A fixed, reviewed indefinite integration question with a complete worked solution and method explanation.

Standard4 marksNon-calculator

Question

Find ∫(6x² − 4x + 3) dx.

Show the worked solution

Worked solution

  1. ∫6x² dx = 6x³/3 = 2x³

    Increase the power from 2 to 3, then divide by 3.

  2. ∫−4x dx = −4x²/2 = −2x²

    Increase the power from 1 to 2, then divide by 2.

  3. ∫3 dx = 3x

  4. ∫(6x² − 4x + 3) dx = 2x³ − 2x² + 3x + C

    Combine the integrated terms and include the constant of integration.

Final answer: 2x³ − 2x² + 3x + C

Why this method works

Indefinite integration reverses differentiation. Integrate each term separately by increasing its power by 1 and dividing by the new power, then add C.

Check your work

Differentiate 2x³ − 2x² + 3x + C. The result is 6x² − 4x + 3, which matches the integrand

Common mistakes

  • Dividing by the old power instead of the new power.
  • Treating the constant 3 as zero.
  • Leaving out + C.