With the substitution t = x^2, what does ∫₀^2 xe^x^2 dx become?

Definite Integrals & Area · Question 23 of 71

Question 23

MediumOriginal question

With the substitution $t = x^2$ , what does $\int_0^2 xe^{x^2}\,dx$ become?

\frac{1}{2}\int_0^4 e^t\,dt

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\frac{1}{2}\int_0^2 e^t\,dt

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2\int_0^4 e^t\,dt

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\int_0^2 e^t\,dt

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Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$dt = 2x\,dx$ , so $x\,dx = \frac{dt}{2}$ .
The limits change too: $x = 0$ gives $t = 0$ and $x = 2$ gives $t = 4$ . So the integral is $\frac{1}{2}\int_0^4 e^t\,dt$ .

Worth noting

Keeping the old limits 0 and 2 for $t$ is the usual slip: they belong to $x$ .

Ten questions from Definite Integrals & Area, at your own pace.

Ten questions from Definite Integrals & Area, at your own pace.