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Question

Question: \[\int_{0}^{1}{e^{2\text{In}x}dx} =\]...

∫01e2Inxdx=\int_{0}^{1}{e^{2\text{In}x}dx} =

A

0

B

12\frac{1}{2}

C

13\frac{1}{3}

D

14\frac{1}{4}

Answer

13\frac{1}{3}

Explanation

Solution

∫01e2log⁡xdx=∫01elog⁡x2dx=∫01x2dx=[x33]01=13\int_{0}^{1}{e^{2\log x}dx = \int_{0}^{1}e^{\log x^{2}}dx = \int_{0}^{1}{x^{2}dx = \left\lbrack \frac{x^{3}}{3} \right\rbrack_{0}^{1} = \frac{1}{3}}}.