The given integral is:

$I = \int_{-\infty}^{1} e^{\frac{-x^2}{2}} \cdot \frac{1}{\sqrt{6.28}} \, dx$

Step 1: Factor out the constant

The term $\frac{1}{\sqrt{6.28}}$ is constant, so we can take it outside the integral:

$I = \frac{1}{\sqrt{6.28}} \int_{-\infty}^{1} e^{\frac{-x^2}{2}} \, dx$

Step 2: Recognize the Gaussian integral

The function $e^{\frac{-x^2}{2}}$ is part of a Gaussian function, which has a known standard result over the entire real line:

$\int_{-\infty}^{\infty} e^{\frac{-x^2}{2}} \, dx = \sqrt{2\pi}$

However, since the upper limit of the given integral is 1 (instead of infinity), we need to evaluate this incomplete Gaussian integral.

Step 3: Use error function

The integral of the Gaussian function over a finite range is related to the error function $\text{erf}(x)$. Specifically:

$\int_{-\infty}^{a} e^{\frac{-x^2}{2}} \, dx = \sqrt{\frac{\pi}{2}} \cdot \text{erf}\left( \frac{a}{\sqrt{2}} \right)$

In our case, $a = 1$, so:

$\int_{-\infty}^{1} e^{\frac{-x^2}{2}} \, dx = \sqrt{\frac{\pi}{2}} \cdot \text{erf}\left( \frac{1}{\sqrt{2}} \right)$

Step 4: Approximate the error function value

Using a standard approximation:

$\text{erf}\left( \frac{1}{\sqrt{2}} \right) \approx 0.6827$

Thus, the integral becomes:

$\int_{-\infty}^{1} e^{\frac{-x^2}{2}} \, dx \approx \sqrt{\frac{\pi}{2}} \cdot 0.6827$

Since $\sqrt{\frac{\pi}{2}} \approx 1.2533$, we get:

$\int_{-\infty}^{1} e^{\frac{-x^2}{2}} \, dx \approx 1.2533 \times 0.6827 \approx 0.856$

Step 5: Multiply by the constant factor

Now, multiply this result by $\frac{1}{\sqrt{6.28}}$:

$I \approx \frac{1}{\sqrt{6.28}} \times 0.856$

Since $\frac{1}{\sqrt{6.28}} \approx 0.3989$, we get:

$I \approx 0.3989 \times 0.856 \approx 0.3416$

Final Answer:

$I \approx 0.3416$

Would you like further details on any step?

Here are 5 related questions for further exploration:

1. How is the error function $\text{erf}(x)$ derived?
2. What is the significance of Gaussian integrals in statistics and physics?
3. Can the integral be solved using numerical methods, and how?
4. How does the incomplete Gaussian integral relate to normal distribution?
5. What are some common approximations used for the error function?

Tip: The error function, $\text{erf}(x)$, often appears in problems related to normal distributions and is closely tied to cumulative probabilities.