To solve the integral \( I = \int_{0}^{\infty} e^{-x^2} \, dx \) and show that it equals \( \frac{\sqrt{\pi}}{2} \), we can follow these steps:
### Step 1: Recognize the integral
The integral \( I = \int_{0}^{\infty} e^{-x^2} \, dx \) is a well-known Gaussian integral.
**Hint:** This integral is often encountered in probability and statistics, particularly in relation to the normal distribution.
### Step 2: Use substitution
To evaluate the integral, we can use a substitution. Let's set \( x = \frac{t}{\sqrt{2}} \). Then, we differentiate to find \( dx \):
\[
dx = \frac{1}{\sqrt{2}} \, dt
\]
**Hint:** When performing substitutions, remember to change the limits of integration accordingly.
### Step 3: Change the limits of integration
When \( x = 0 \), \( t = 0 \). When \( x \to \infty \), \( t \to \infty \). Thus, the limits remain the same.
**Hint:** Always check how the limits of integration change when you perform a substitution.
### Step 4: Substitute into the integral
Now substitute \( x \) and \( dx \) into the integral:
\[
I = \int_{0}^{\infty} e^{-\left(\frac{t}{\sqrt{2}}\right)^2} \cdot \frac{1}{\sqrt{2}} \, dt
\]
This simplifies to:
\[
I = \frac{1}{\sqrt{2}} \int_{0}^{\infty} e^{-\frac{t^2}{2}} \, dt
\]
**Hint:** Simplifying the exponent can make the integral easier to evaluate.
### Step 5: Recognize the new integral
The integral \( \int_{0}^{\infty} e^{-\frac{t^2}{2}} \, dt \) can be evaluated using the known result for the Gaussian integral:
\[
\int_{-\infty}^{\infty} e^{-u^2} \, du = \sqrt{\pi}
\]
However, we need to adjust for the factor of \( \frac{1}{2} \) in the exponent.
**Hint:** The integral from \( 0 \) to \( \infty \) is half of the integral from \( -\infty \) to \( \infty \).
### Step 6: Evaluate the integral
We can relate our integral to the known result:
\[
\int_{0}^{\infty} e^{-\frac{t^2}{2}} \, dt = \sqrt{2} \cdot \frac{\sqrt{\pi}}{2}
\]
Thus, substituting back, we have:
\[
I = \frac{1}{\sqrt{2}} \cdot \sqrt{2} \cdot \frac{\sqrt{\pi}}{2} = \frac{\sqrt{\pi}}{2}
\]
**Hint:** Always keep track of constants when substituting and simplifying.
### Conclusion
Therefore, we conclude that:
\[
\int_{0}^{\infty} e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2}
\]
