Simpson Harrington
04/23/2023 · High School
\( \int e ^ { - \frac { x ^ { 2 } } { 2 } } d x \)
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Tutor-Verified Answer
Step-by-step Solution
The integral \( \int e^{-\frac{x^2}{2}} \, dx \) does not have a closed-form expression in terms of elementary functions. However, it is related to the Gaussian integral and the error function.
To express the integral in terms of the error function, we can use the following relationship:
\[
\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt
\]
To relate this to our integral, we can make a substitution. Let:
\[
u = \frac{x}{\sqrt{2}} \quad \Rightarrow \quad x = \sqrt{2}u \quad \Rightarrow \quad dx = \sqrt{2} \, du
\]
Now, substituting into the integral:
\[
\int e^{-\frac{x^2}{2}} \, dx = \int e^{-u^2} \sqrt{2} \, du = \sqrt{2} \int e^{-u^2} \, du
\]
The integral \( \int e^{-u^2} \, du \) does not have a closed form, but we can express it in terms of the error function:
\[
\int e^{-u^2} \, du = \frac{\sqrt{\pi}}{2} \text{erf}(u) + C
\]
Thus, substituting back for \( u \):
\[
\int e^{-\frac{x^2}{2}} \, dx = \sqrt{2} \left( \frac{\sqrt{\pi}}{2} \text{erf}\left(\frac{x}{\sqrt{2}}\right) \right) + C
\]
This simplifies to:
\[
\int e^{-\frac{x^2}{2}} \, dx = \frac{\sqrt{2\pi}}{2} \text{erf}\left(\frac{x}{\sqrt{2}}\right) + C
\]
where \( C \) is the constant of integration.
Quick Answer
The integral \( \int e^{-\frac{x^2}{2}} \, dx \) is equal to \( \frac{\sqrt{2\pi}}{2} \text{erf}\left(\frac{x}{\sqrt{2}}\right) + C \), where \( C \) is the constant of integration.
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