Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 8

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- Solution 8

Solution

Use integration by substitution with

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}}

, confirming the antiderivative. The real constant of integration

C

is arbitrary.

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