To evaluate the integral

$\int_{1}^{\infty} \frac{1}{x^2 + \sin^2(x)} \, dx,$

we can analyze its convergence. Let's examine the integrand $\frac{1}{x^2 + \sin^2(x)}$.

Step 1: Behavior at infinity

As $x$ approaches infinity, $\sin^2(x)$ oscillates between 0 and 1. Therefore, $x^2 + \sin^2(x)$ behaves approximately like $x^2$. Thus, for large $x$:

$\frac{1}{x^2 + \sin^2(x)} \approx \frac{1}{x^2}.$

The integral $\int_{1}^{\infty} \frac{1}{x^2} \, dx$ is a convergent integral:

$\int_{1}^{\infty} \frac{1}{x^2} \, dx = \left. -\frac{1}{x} \right|_{1}^{\infty} = 1.$

Step 2: Behavior near 1

Near $x = 1$, $\frac{1}{x^2 + \sin^2(x)}$ is continuous and finite because both $x^2$ and $\sin^2(x)$ are bounded and nonzero.

Conclusion

Since the integrand behaves similarly to $\frac{1}{x^2}$ at infinity, and is continuous and bounded near 1, the integral

$\int_{1}^{\infty} \frac{1}{x^2 + \sin^2(x)} \, dx$

is convergent.

Thus, we conclude that the integral converges, although finding the exact value of this improper integral analytically is quite challenging and typically requires advanced techniques or numerical methods.

Would you like further details on any specific part of this explanation?

Here are five questions you might be interested in asking:

1. How do you evaluate improper integrals in general?
2. What are some common techniques for evaluating challenging integrals?
3. Can you explain more about the convergence of improper integrals?
4. How does one typically handle integrals involving trigonometric functions?
5. What are some practical applications of improper integrals?

Tip: When dealing with improper integrals, always first check if the integrand behaves similarly to a known integrable function to assess convergence.