Ortega Flynn
10/25/2023 · Middle School
\( \left. \begin{array} { l } { d v = \int x e ^ { - \frac { x ^ { 2 } } { 2 } } d x } \\ { v = } \end{array} \right. \)
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Tutor-Verified Answer
Step-by-step Solution
To find \( v \) from the equation \( dv = \int x e^{-\frac{x^2}{2}} \, dx \), we need to evaluate the integral on the right-hand side.
We can use integration by substitution. Let:
\[
u = -\frac{x^2}{2} \implies du = -x \, dx \implies dx = -\frac{du}{x}
\]
However, we can also directly recognize that the integral can be solved using the properties of the Gaussian integral. The integral we have is:
\[
\int x e^{-\frac{x^2}{2}} \, dx
\]
This integral can be solved by recognizing that the derivative of \( e^{-\frac{x^2}{2}} \) is:
\[
\frac{d}{dx} e^{-\frac{x^2}{2}} = -x e^{-\frac{x^2}{2}}
\]
Thus, we can rewrite our integral as:
\[
\int x e^{-\frac{x^2}{2}} \, dx = -e^{-\frac{x^2}{2}} + C
\]
where \( C \) is the constant of integration.
Therefore, we have:
\[
v = -e^{-\frac{x^2}{2}} + C
\]
So the final answer is:
\[
v = -e^{-\frac{x^2}{2}} + C
\]
Quick Answer
\( v = -e^{-\frac{x^2}{2}} + C \)
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