Find ∫ (3x + 5)/√(x² + 4x + 10) 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 = ∫ (3x + 5)/√(x² + 4x + 10) dx. To find: I.
2 · Write the rules.
- px + q = A × d/dx (x² + 4x + 10) + B
- ∫ (2x + 4)/√(x² + 4x + 10) dx = 2√(x² + 4x + 10)
- ∫ dt/√(t² + a²) = log|t + √(t² + a²)| + C
3 · Find A and B. 3x + 5 = A(2x + 4) + B. Coefficient of x: 2A = 3, so A = 3/2. Constants: 4A + B = 5, so B = −1.
4 · Integrate the first part. (3/2) ∫ (2x + 4)/√(x² + 4x + 10) dx = (3/2) × 2√(x² + 4x + 10) = 3√(x² + 4x + 10).
5 · Integrate the second part. x² + 4x + 10 = (x + 2)² + (√6)². So −∫ dx/√((x + 2)² + (√6)²) = −log|x + 2 + √(x² + 4x + 10)|.
6 · Check and box the answer. I = 3√(x² + 4x + 10) − log|x + 2 + √(x² + 4x + 10)| + C.

Scoring tip: Find A from the x terms first, then B from the constants. One wrong sign in B spoils the whole answer.