Find ∫ x²/((x² + 4)(x² + 9)) dx.

1. 1 · Write what is given and what is to be found. I = ∫ x²/((x² + 4)(x² + 9)) dx. To find: I.
2. 2 · Write the rules.
   • Partial fractions for a proper rational function.
   • ∫ dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C.
3. 3 · Put x² = y, only to find the partial fractions. y/((y + 4)(y + 9)) = A/(y + 4) + B/(y + 9), so y = A(y + 9) + B(y + 4).
4. 4 · Find the constants.
   • Put y = −4: −4 = 5A, so A = −4/5.
   • Put y = −9: −9 = −5B, so B = 9/5.
5. 5 · Rewrite in x. x²/((x² + 4)(x² + 9)) = −(4/5) × 1/(x² + 4) + (9/5) × 1/(x² + 9).
6. 6 · Integrate each part. I = −(4/5)(1/2) tan⁻¹(x/2) + (9/5)(1/3) tan⁻¹(x/3) + C.
7. 7 · Check and box the answer. I = −(2/5) tan⁻¹(x/2) + (3/5) tan⁻¹(x/3) + C. Check: −(4/5)/(x² + 4) + (9/5)/(x² + 9) = x²/((x² + 4)(x² + 9)).

Scoring tip: Say clearly that x² = y is used only to find A and B. Integrate in x, not in y.