If `int_(0)^(oo) e^(-x^(2))dx=sqrt((pi)/(2))"then"int_(0)^(oo) e^(-ax^(2)) dx, a gt0`, s

Options

A. `(sqrt(pi))/(2)`
B. `(sqrt(pi))/(2a)`
C. `2(sqrt(pi))/(a)`
D. `(1)/(2)(sqrt((pi)/(a)))`

Answer

To solve the integral \( \int_{0}^{\infty} e^{-ax^2} \, dx \) where \( a > 0 \), we can use a substitution method based on the given integral \( \int_{0}^{\infty} e^{-x^2} \, dx = \sqrt{\frac{\pi}{2}} \).

### Step-by-Step Solution:

1. **Substitution**:
Let \( ax^2 = t \). Then, we differentiate to find \( dx \):
\[ 2ax \, dx = dt \quad \Rightarrow \quad dx = \frac{dt}{2ax} \]
Since \( x = \sqrt{\frac{t}{a}} \), we can substitute this into the expression for \( dx \):
\[ dx = \frac{dt}{2a \sqrt{\frac{t}{a}}} = \frac{dt}{2\sqrt{a} \sqrt{t}} \]

2. **Change the limits**:
When \( x = 0 \), \( t = 0 \) and when \( x \to \infty \), \( t \to \infty \). Thus, the limits of integration remain from \( 0 \) to \( \infty \).

3. **Substituting into the integral**:
Now we can rewrite the integral:
\[ \int_{0}^{\infty} e^{-ax^2} \, dx = \int_{0}^{\infty} e^{-t} \cdot \frac{dt}{2\sqrt{a} \sqrt{t}} \]
This simplifies to:
\[ = \frac{1}{2\sqrt{a}} \int_{0}^{\infty} e^{-t} \, dt \]

4. **Evaluate the integral**:
The integral \( \int_{0}^{\infty} e^{-t} \, dt = 1 \). Therefore:
\[ \int_{0}^{\infty} e^{-ax^2} \, dx = \frac{1}{2\sqrt{a}} \]

5. **Final expression**:
We know from the problem statement that \( \int_{0}^{\infty} e^{-x^2} \, dx = \sqrt{\frac{\pi}{2}} \). We can relate this to our result:
\[ \int_{0}^{\infty} e^{-ax^2} \, dx = \frac{\sqrt{\pi}}{2\sqrt{a}} \]

### Conclusion:
Thus, the final result is:
\[ \int_{0}^{\infty} e^{-ax^2} \, dx = \frac{\sqrt{\pi}}{2\sqrt{a}} \]