Mathematics

# The value  of the definite integral, $I = \int _{ 0 }^{ \sqrt { 10 } }{ \dfrac { x }{ { e }^{ { x }^{ 2 } } } } dx$ is equal to

##### SOLUTION
$I = \displaystyle \int_0^{\sqrt{10}} \frac{x}{e^{x^2}}dx \quad x^2 = t$

$\displaystyle \Rightarrow 2x . dx = dt \Rightarrow x \, dx = \frac{dt}{2}$

$I = \displaystyle \int_0^{\sqrt{10}} \frac{dt}{2\times e^t} =\frac{1}{2}\int_0^{\sqrt{10}} e^{-t} \, dt$

$I \displaystyle =\frac{-1}{2}[e^{-t}]_0^{\sqrt{10}} = \frac{-1}{2}[e^{-x^2}]_0^{\sqrt{10}}$

$I \displaystyle =\frac{-1}{2} \left[e^{-10}- e^0\right] = \frac{-1}{2}\left[[\frac{1}{e^2}]^5 - 1\right]$

$\displaystyle = 0.499$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

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