The integral provided is:

$\int_0^\infty e^{-x^3} \, dx$

This is a non-elementary integral, meaning it cannot be expressed in terms of elementary functions. However, it can be evaluated using special functions, specifically the Gamma function or substitutions leading to its representation.

Step-by-step solution:

1. Substitution: Let $x^3 = t$, so that $x = t^{1/3}$ and $dx = \frac{1}{3} t^{-2/3} dt$.

   Changing the limits:

   • When $x = 0$, $t = 0$.
   • When $x = \infty$, $t = \infty$.

   The integral becomes: $\int_0^\infty e^{-x^3} \, dx = \int_0^\infty e^{-t} \cdot \frac{1}{3} t^{-2/3} \, dt$

2. Rewriting the integral: Factor out the constant $\frac{1}{3}$: $\int_0^\infty e^{-x^3} \, dx = \frac{1}{3} \int_0^\infty t^{-2/3} e^{-t} \, dt$

3. Recognizing the Gamma function: The Gamma function is defined as: $\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt$

   Comparing with the integral above, we see that $z - 1 = -\frac{2}{3}$, so $z = \frac{1}{3}$.

   Therefore: $\int_0^\infty e^{-x^3} \, dx = \frac{1}{3} \Gamma\left(\frac{1}{3}\right)$

4. Final result: The value of the integral is: $\int_0^\infty e^{-x^3} \, dx = \frac{1}{3} \Gamma\left(\frac{1}{3}\right)$

   The Gamma function $\Gamma\left(\frac{1}{3}\right)$ is a well-known constant that can be approximated numerically, but it does not simplify further in elementary terms.

Numerical Approximation:

Using a numerical approximation for $\Gamma\left(\frac{1}{3}\right)$: $\Gamma\left(\frac{1}{3}\right) \approx 2.67894$

Thus: $\int_0^\infty e^{-x^3} \, dx \approx \frac{1}{3} \cdot 2.67894 \approx 0.89298$

Would you like further details on Gamma functions or a more detailed numerical explanation?

5 Related Questions:

1. What is the Gamma function and how is it defined for various values?
2. How can this integral be solved using alternative substitutions?
3. Why is this integral considered non-elementary, and what are some other examples?
4. How is the Gamma function related to factorials for integer arguments?
5. Can this integral be solved using series expansions for $e^{-x^3}$?

Tip:

The Gamma function generalizes the factorial function for non-integer values and is highly useful in advanced calculus and probability theory!