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Integration by substitution sheet
3-mark answer frame

Find ∫ (x + 1)(x + log x)³/x dx.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found. I = ∫ (x + 1)(x + log x)³/x dx. To find: I.
  2. 2 · Write the rule. Substitution: if t = g(x), then dt = g′(x) dx. Here d/dx (x + log x) = 1 + 1/x = (x + 1)/x, which is a factor of the integrand.
  3. 3 · Substitute.
    • Put t = x + log x.
    • dt = (x + 1)/x dx.
    • I = ∫ t³ dt.
  4. 4 · Integrate in t. I = t⁴/4 + C.
  5. 5 · Put back x. I = (x + log x)⁴/4 + C.
  6. 6 · Check and box the answer. d/dx [(x + log x)⁴/4] = (x + log x)³ × (x + 1)/x, the integrand. So I = (x + log x)⁴/4 + C.

Scoring tip: Show the line dt = (x + 1)/x dx. It carries the method mark even if the final answer slips.