To evaluate the integral

$I = \int_{-\infty}^{-1} \left(\frac{x^4}{1 + x^6}\right)^2 dx$

we start by simplifying the integrand:

$I = \int_{-\infty}^{-1} \frac{x^8}{(1 + x^6)^2} dx$

Next, we apply the substitution $u = -x$, so $du = -dx$. The limits of integration change as follows:

• When $x = -\infty$, $u = \infty$
• When $x = -1$, $u = 1$

Thus, the integral becomes:

$I = \int_{\infty}^{1} \frac{(-u)^8}{(1 + (-u)^6)^2} (-du) = \int_1^{\infty} \frac{u^8}{(1 + u^6)^2} du$

We can see that this integral is symmetric around $x = 0$. The original integral from $-\infty$ to $-1$ is equal to the integral from 1 to $\infty$:

$I = \int_{1}^{\infty} \frac{u^8}{(1 + u^6)^2} du$

Thus, the original integral over the symmetric interval is:

$I = 2 \times \int_{1}^{\infty} \frac{x^8}{(1 + x^6)^2} dx$

Now, let's evaluate this integral. At large values of $x$, the integrand behaves as $x^{-4}$, ensuring that the integral converges. However, the exact analytical evaluation is complex, and typically, this integral would be solved using numerical methods.

Since the integral is symmetric and converges, it has a finite value, though finding an explicit form may require advanced techniques or numerical computation. A standard result or reference might give the exact value, but based on the form, this integral does not simplify nicely into elementary functions.

Would you like to explore this integral further numerically or review similar integrals?

Related Questions:

1. How do you apply substitution in integrals involving symmetric limits?
2. What are the conditions for the convergence of improper integrals?
3. Can integrals like these be solved using contour integration in the complex plane?
4. How does the behavior of the integrand at infinity affect the convergence of an integral?
5. What numerical methods are typically used to approximate integrals like this?

Tip:

When dealing with improper integrals, checking the behavior of the integrand at the boundaries is crucial to determine whether the integral converges.