Find ∫ √(x² − 4x − 5) dx.

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

1 · Write what is given and what is to be found. I = ∫ √(x² − 4x − 5) dx. To find: I.
2 · Write the formula. ∫ √(t² − a²) dt = (t/2)√(t² − a²) − (a²/2) log|t + √(t² − a²)| + C.
3 · Complete the square. x² − 4x − 5 = (x − 2)² − 4 − 5 = (x − 2)² − 3².
4 · Substitute. Put t = x − 2, dt = dx, a = 3. I = ∫ √(t² − 3²) dt.
5 · Apply the formula. I = (t/2)√(t² − 9) − (9/2) log|t + √(t² − 9)| + C.
6 · Check and box the answer. Put back t = x − 2: I = ((x − 2)/2)√(x² − 4x − 5) − (9/2) log|x − 2 + √(x² − 4x − 5)| + C.

Scoring tip: Write which of the three results you are using. The sign before a²/2 is where most marks are lost.