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
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