A-level · Integration

Integration by partsExam-style questions

Integrating a product of two unrelated factors by reversing the product rule. Where the formula comes from; choosing which factor to differentiate, so the integral left over is easier; integrating ln x by writing it as ln x × 1; applying parts twice when a power of x needs two passes; and definite integrals by parts, with exact answers.

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

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    Find ∫x ex dx\displaystyle\int x\,e^x\,dx.
    Level 2 · 3 marksTry it
  2. 2
    Find ∫xcos⁡x dx\displaystyle\int x\cos x\,dx.
    Level 2 · 3 marksTry it
  3. 3
    Find ∫ln⁡x dx\displaystyle\int \ln x\,dx.
    Level 3 · 3 marksTry it
  4. 4
    Find ∫xsin⁡2x dx\displaystyle\int x\sin 2x\,dx.
    Level 3 · 3 marksTry it
  5. 5
    Evaluate ∫01x e2x dx\displaystyle\int_0^1 x\,e^{2x}\,dx, giving your answer in terms of ee.
    Level 3 · 4 marksTry it

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