 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/17/b/i/solution

x = 2 sin θ

with

- π /2 < θ < π /2

, so

d x = 2 cos θ d θ

and

4 - x^{2} = 2 cos θ

\int \frac{d x}{4 - x ^{2}} = arcsin \frac{x}{2} + C, - 2 < x < 2.

This uses the principal inverse sine; the chosen angle interval keeps the square root positive.

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