We need to evaluate the following product of two integrals:

$I = \left( \int_{-\infty}^{\infty} x^2 e^{-x^4} \,dx \right) \times \left( \int_0^{\infty} e^{-x^4} \,dx \right)$

Step 1: Evaluating $I_1 = \int_{-\infty}^{\infty} x^2 e^{-x^4} \,dx$

Since the integrand is an even function ($x^2 e^{-x^4}$ is symmetric about $x = 0$), we can express it as:

$I_1 = 2 \int_0^{\infty} x^2 e^{-x^4} \,dx$

Using the substitution:

$u = x^4 \Rightarrow du = 4x^3 \,dx \Rightarrow dx = \frac{du}{4x^3}$

Rewriting $x^2$ in terms of $u$:

$x^2 = u^{1/2}, \quad x^3 = u^{3/4}$

Thus, the integral transforms into:

$I_1 = 2 \int_0^{\infty} u^{1/2} e^{-u} \cdot \frac{du}{4u^{3/4}}$

$= \frac{1}{2} \int_0^{\infty} u^{(1/2 - 3/4)} e^{-u} \,du$

$= \frac{1}{2} \int_0^{\infty} u^{-1/4} e^{-u} \,du$

Recognizing this as the Gamma function $\Gamma(s) = \int_0^{\infty} u^{s-1} e^{-u} \,du$, we set $s - 1 = -\frac{1}{4}$ so that $s = \frac{3}{4}$:

$I_1 = \frac{1}{2} \Gamma(3/4)$

Step 2: Evaluating $I_2 = \int_0^{\infty} e^{-x^4} \,dx$

Using the substitution $v = x^4 \Rightarrow dv = 4x^3 dx$, we get:

$I_2 = \int_0^{\infty} e^{-v} \cdot \frac{dv}{4v^{3/4}}$

$= \frac{1}{4} \int_0^{\infty} v^{-3/4} e^{-v} \,dv$

This is again a Gamma function $\Gamma(s)$ with $s - 1 = -\frac{3}{4}$ so that $s = \frac{1}{4}$:

$I_2 = \frac{1}{4} \Gamma(1/4)$

Step 3: Computing the Product $I = I_1 \times I_2$

$I = \left(\frac{1}{2} \Gamma(3/4) \right) \times \left(\frac{1}{4} \Gamma(1/4) \right)$

$= \frac{1}{8} \Gamma(3/4) \Gamma(1/4)$

Using the Gamma function identity:

$\Gamma(x) \Gamma(1-x) = \frac{\pi}{\sin(\pi x)}$

for $x = \frac{1}{4}$, we get:

$\Gamma(1/4) \Gamma(3/4) = \frac{\pi}{\sin(\pi/4)} = \frac{\pi}{\frac{\sqrt{2}}{2}} = \pi \sqrt{2}$

Thus,

$I = \frac{1}{8} \pi \sqrt{2} = \frac{\pi \sqrt{2}}{8}$

Final Answer:

$\frac{\pi \sqrt{2}}{8}$

Would you like further clarification on any step?

Related Questions:

1. How do you evaluate Gaussian integrals of the form $\int e^{-x^n} dx$?
2. What is the Gamma function and how is it related to factorials?
3. How can substitution methods simplify integral evaluations?
4. What are common techniques for solving improper integrals?
5. How does the sine-Gamma identity help in computing integrals?

Tip:

The Gamma function generalizes the factorial, and its special values are often useful in evaluating definite integrals.