We are tasked to evaluate the integral:

$\int_{0}^{\infty} 3^{-4x^2} \, dx$

Solution:

Let us rewrite the integrand using properties of exponents:

$3^{-4x^2} = e^{-4x^2 \ln 3}$

The integral becomes:

$\int_{0}^{\infty} e^{-4x^2 \ln 3} \, dx$

This is a Gaussian integral of the form:

$\int_{0}^{\infty} e^{-a x^2} \, dx = \frac{1}{2} \sqrt{\frac{\pi}{a}}, \quad \text{where } a > 0.$

Here, $a = 4 \ln 3$. Applying the formula:

$\int_{0}^{\infty} e^{-4x^2 \ln 3} \, dx = \frac{1}{2} \sqrt{\frac{\pi}{4 \ln 3}}$

Simplify:

$\int_{0}^{\infty} 3^{-4x^2} \, dx = \frac{1}{2} \cdot \frac{\sqrt{\pi}}{\sqrt{4 \ln 3}} = \frac{\sqrt{\pi}}{2 \sqrt{4 \ln 3}} = \frac{\sqrt{\pi}}{4 \sqrt{\ln 3}}.$

Final Answer:

$\int_{0}^{\infty} 3^{-4x^2} \, dx = \frac{\sqrt{\pi}}{4 \sqrt{\ln 3}}.$

Would you like me to expand on any step?

Related Questions:

1. How is the Gaussian integral formula derived?
2. Why is $3^{-4x^2}$ rewritten using the natural logarithm $\ln 3$?
3. How does the property of exponents simplify complex expressions in integrals?
4. What happens if the limits of integration change (e.g., from 0 to 1)?
5. How can this integral be computed numerically?

Tip:

For integrals involving $e^{-ax^2}$, always consider the Gaussian integral formula to simplify computations.