Evaluate ∫ sin 2x/(1 + cos² x) dx from 0 to π/2.

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 = ∫ sin 2x/(1 + cos² x) dx from 0 to π/2. To find: the value of I.
2 · Write the rule. Second fundamental theorem: ∫ f(x) dx from a to b = F(b) − F(a). With a substitution, change the limits too.
3 · Substitute.
- Put t = 1 + cos² x.
- dt = 2 cos x (−sin x) dx = −sin 2x dx.
4 · Change the limits. x = 0 gives t = 1 + 1 = 2. x = π/2 gives t = 1 + 0 = 1.
5 · Integrate. I = ∫ (−1/t) dt from 2 to 1 = ∫ (1/t) dt from 1 to 2 = [log t] from 1 to 2.
6 · Check and box the answer. I = log 2 − log 1 = log 2.

Scoring tip: Write the new limits on a separate line. Examiners look for them.