A-level · Integration

Indefinite integration (reversing differentiation)Exam-style questions

Integration as differentiation run backwards — finding a function from its gradient. The power rule (raise the power, divide by the new power) and why it undoes differentiation; numbers in front and constant terms; rewriting roots and fractions as powers; why every answer ends in + C, and how a point on the curve fixes it; multiplying out brackets and splitting fractions before you integrate; and checking an answer by differentiating it.

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11 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4
  1. 1
    The expression (3x−1)(x+2)2(3x - 1)(x + 2)^2 can be written in the form

    3x3+ax2+bx−43x^3 + ax^2 + bx - 4

    where aa and bb are integers.
    (a) Find the value of aa and the value of bb.

    (2 marks)
    (b) Hence find

    ∫(3x−1)(x+2)2xx dxx>0\int \frac{(3x - 1)(x + 2)^2}{x\sqrt{x}}\,\mathrm{d}x \qquad x > 0

    giving each term in its simplest form.

    (5 marks)
    Level 4 · 7 marksTry it