 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/5/solution

For the indefinite integral, substitute

u = x^{2}

, so

x^{3} d x = u d u /2

\int x^{3} cos (x^{2}) d x = \frac{1}{2} \int u cos u d u = \frac{1}{2} [x^{2} sin (x^{2}) + cos (x^{2})] + C .

For the definite integral, the original PDF's upper limit is

π /2

, so the corresponding upper value of

u

π /2

. Therefore

\int_{0}^{π /2} x^{3} cos (x^{2}) d x = \frac{1}{2} [\frac{π}{2} sin \frac{π}{2} + cos \frac{π}{2} - 1] = \frac{π - 2}{4} .

Both requested integrals are thus evaluated with the same antiderivative.

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