To solve the integral

$\int_0^\infty e^{-x^2} x^9 \, dx = 12,$

we will proceed step by step, leveraging the Gamma function properties and substitutions.

Step 1: General Gamma Function Relation

The Gamma function is defined as:

$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt,$

and a key property of definite integrals involving $e^{-x^2}$ is:

$\int_0^\infty x^{2n} e^{-x^2} \, dx = \frac{\Gamma\left(n + \frac{1}{2}\right)}{2}.$

Here, $x^{2n}$ indicates even powers of $x$. For odd powers $x^{2n+1}$, the integral equals 0 because the function would be odd over symmetric limits.

Step 2: Relate the Given Integral to the Gamma Function

In the given problem, we have $x^9 e^{-x^2}$, where $x^9 = x^{2 \cdot 4 + 1} x$. Thus, we decompose this as follows:

Let $x = t^{1/2}$, or $t = x^2$. This implies $dx = \frac{1}{2} t^{-1/2} dt$.

Substituting $t$ into the integral:

$\int_0^\infty x^9 e^{-x^2} \, dx = \int_0^\infty \left(t^{1/2}\right)^9 e^{-t} \cdot \frac{1}{2} t^{-1/2} \, dt.$

Simplify:

$\int_0^\infty x^9 e^{-x^2} \, dx = \frac{1}{2} \int_0^\infty t^{9/2} t^{-1/2} e^{-t} \, dt.$

$= \frac{1}{2} \int_0^\infty t^{(9/2) + (-1/2)} e^{-t} \, dt.$

$= \frac{1}{2} \int_0^\infty t^{4} e^{-t} \, dt.$

Step 3: Use the Gamma Function

The remaining integral is now in the form of the Gamma function:

$\int_0^\infty t^{4} e^{-t} \, dt = \Gamma(4 + 1) = \Gamma(5).$

The Gamma function satisfies the property:

$\Gamma(n) = (n-1)!.$

Thus:

$\Gamma(5) = 4! = 24.$

Therefore:

$\int_0^\infty x^9 e^{-x^2} \, dx = \frac{1}{2} \cdot 24 = 12.$

Final Answer:

$\int_0^\infty e^{-x^2} x^9 \, dx = 12.$

This confirms the result.

Would you like additional details on the Gamma function or related integrals?

5 Related Questions:

1. What is the Gamma function and how is it related to factorials?
2. How can you compute integrals of the form $\int_0^\infty x^{n} e^{-x^2} dx$ for other values of $n$?
3. What substitution rules are commonly used to solve these types of integrals?
4. Can we solve this problem using numerical integration instead of analytical methods?
5. How does the property of symmetry affect integrals involving even and odd powers of $x$?

Tip:

Always check if a given integral can be expressed in terms of standard functions like Gamma or Beta functions to simplify the evaluation process.