Evaluate ∫ x sin x/(3 + cos² x) dx from 0 to π.

1. 1 · Write what is given and what is to be found. I = ∫ x sin x/(3 + cos² x) dx from 0 to π. To find: the value of I.
2. 2 · Write the property. ∫ f(x) dx from 0 to a = ∫ f(a − x) dx from 0 to a, with a = π.
3. 3 · Replace x by π − x. sin(π − x) = sin x and cos²(π − x) = cos² x. So I = ∫ (π − x) sin x/(3 + cos² x) dx from 0 to π.
4. 4 · Add the two forms. 2I = π ∫ sin x/(3 + cos² x) dx from 0 to π. The x has cancelled.
5. 5 · Substitute.
   • Put t = cos x, dt = −sin x dx.
   • x = 0 gives t = 1; x = π gives t = −1.
   • 2I = π ∫ dt/(3 + t²) from −1 to 1.
6. 6 · Integrate. 2I = π × (1/√3)[tan⁻¹(t/√3)] from −1 to 1 = (π/√3)(π/6 + π/6) = π²/(3√3).
7. 7 · Check and box the answer. I = π²/(6√3).

Scoring tip: State the property in words before you use it. The step where x cancels after adding is the key method mark.