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Special integrals: 1/(x² ± a²), 1/(a² − x²) and their square roots sheet
3-mark answer frame

Find ∫ eˣ/(e²ˣ − 9) 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 = ∫ eˣ/(e²ˣ − 9) dx. To find: I.
  2. 2 · Write the formula. ∫ dt/(t² − a²) = (1/2a) log|(t − a)/(t + a)| + C.
  3. 3 · Substitute.
    • Put t = eˣ. Then dt = eˣ dx.
    • e²ˣ = (eˣ)² = t².
    • I = ∫ dt/(t² − 3²).
  4. 4 · Apply the formula with a = 3. I = (1/6) log|(t − 3)/(t + 3)| + C.
  5. 5 · Check and box the answer. Put back t = eˣ: I = (1/6) log|(eˣ − 3)/(eˣ + 3)| + C.

Scoring tip: Write the standard formula before you use it. A mark is given for the correct formula.