Integrate the function frac{x}{e^{x^{2}}}. | vedclass

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(A) To integrate the function $I = \int \frac{x}{e^{x^{2}}} dx$,we use the method of substitution.Let $x^{2} = t$.Differentiating both sides with resp

Vedclass · Accepted Answer

$-\frac{1}{2} e^{-x^{2}} + C$

Vedclass · Answer

(A) To integrate the function $I = \int \frac{x}{e^{x^{2}}} dx$,we use the method of substitution.Let $x^{2} = t$.Differentiating both sides with respect to $x$,we get $2x dx = dt$,which implies $x dx = \frac{1}{2} dt$.Substituting these into the integral:$I = \int \frac{1}{e^{t}} \cdot \frac{1}{2} dt$$I = \frac{1}{2} \int e^{-t} dt$Integrating $e^{-t}$ with respect to $t$ gives $-e^{-t}$:$I = \frac{1}{2} (-e^{-t}) + C$$I = -\frac{1}{2} e^{-t} + C$Substituting back $t = x^{2}$:$I = -\frac{1}{2} e^{-x^{2}} + C = -\frac{1}{2e^{x^{2}}} + C$,where $C$ is an arbitrary constant.

Integrate the function $\frac{x}{e^{x^{2}}}$.

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