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
    Find

    ∫(x2−3)(2+5x2)dx\int \left(x^2 - 3\right)\left(2 + \frac{5}{x^2}\right)\mathrm{d}x

    writing your answer in simplest form.

    (4 marks)
    Level 2 · 4 marksTry it
  2. 2
    Find ∫3x2 dx\displaystyle\int 3x^2 \,\mathrm{d}x.
    Level 2 · 2 marksTry it
  3. 3
    Find

    ∫(8x3−3x2+5x)dx\int \left(8x^3 - \frac{3}{x^2} + 5\sqrt{x}\right)dx

    writing each term in simplest form.

    (4 marks)
    Level 2 · 4 marksTry it
  4. 4
    Find

    ∫6x3−5xx+82x2 dx\int \frac{6x^3 - 5x\sqrt{x} + 8}{2x^2}\,\mathrm{d}x

    writing your answer in simplest form.

    (5 marks)
    Level 3 · 5 marksTry it
  5. 5
    Find ∫(4x3−2x) dx\displaystyle\int (4x^3 - 2x) \,\mathrm{d}x.
    Level 3 · 2 marksTry it

6 more questions on indefinite integration (reversing differentiation)

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