 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/8/solution

u = x^{4}

, so

d u = 4 x^{3} d x

. Then

\int x^{3} e^{- x^{4}} d x = \frac{1}{4} \int e^{- u} d u = - \frac{1}{4} e^{- x^{4}} + C .

Differentiating the answer by the chain rule gives

x^{3} e^{- x^{4}}

C

is arbitrary.

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