Higher Maths worked question

Integrate a cubic polynomial term by term

Practise integrating a polynomial term by term, including the constant of integration, then check your method with a full worked solution.

Standard4 marksNon-calculator

Question

Find ∫(8x3 − 6x2 + 4) dx.

Show the worked solution

Worked solution

  1. ∫8x3 dx = 8x4/4 = 2x4

    Increase the power from 3 to 4, then divide by the new power 4.

  2. ∫−6x2 dx = −6x3/3 = −2x3

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

  3. ∫4 dx = 4x

    A constant integrates to that constant multiplied by x.

  4. ∫(8x3 − 6x2 + 4) dx = 2x4 − 2x3 + 4x + C

    Combine the antiderivatives and include the arbitrary constant.

Final answer: 2x4 − 2x3 + 4x + C

Why this method works

Indefinite integration reverses differentiation. Apply the reverse power rule to each polynomial term separately, integrate the constant as a multiple of x, and finish with the constant of integration.

Check your work

Differentiate 2x4 − 2x3 + 4x + C. The derivative is 8x3 − 6x2 + 4, exactly matching the integrand.

Common mistakes

  • Dividing by the original power instead of the new power.
  • Treating the constant 4 as if it integrates to zero.
  • Leaving out + C from an indefinite integral.