 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/18/b/solution

Evaluate the recurrence boundary terms at zero and

π /2

. For

n > 1

, both

cos x sin^{n - 1} x

and

sin x cos^{n - 1} x

vanish at both endpoints. Therefore the Wallis integrals obey

\int_{0}^{π /2} sin^{n} x d x = \frac{n - 1}{n} \int_{0}^{π /2} sin^{n - 2} x d x,

\int_{0}^{π /2} cos^{n} x d x = \frac{n - 1}{n} \int_{0}^{π /2} cos^{n - 2} x d x .

The starting value for both is

\int_{0}^{π /2} 1 d x = π /2

. Thus

\int_{0}^{π /2} sin^{2} x d x = \int_{0}^{π /2} cos^{2} x d x = \frac{π}{4},

\int_{0}^{π /2} sin^{4} x d x = \int_{0}^{π /2} cos^{4} x d x = \frac{3 π}{16} .

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