 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/13/a/solution

e^{x /2}

to be

2 e^{x /2}

gives

\int x^{2} e^{x /2} d x = 2 x^{2} e^{x /2} - 4 \int x e^{x /2} d x,

and the second application gives

\int x e^{x /2} d x = 2 x e^{x /2} - 4 e^{x /2} + C

. Therefore the required definite integral is

[2 e^{x /2} (x^{2} - 4 x + 8)]_{0}^{1} = 10 e - 16 .

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