The task is to prove that the improper integral

$\int_{1}^{\infty} \frac{\sin x}{x} \, dx$

converges by first applying integration by parts and then using the comparison theorem mentioned in the problem.

Step 1: Integration by Parts

Using the formula for integration by parts:

$\int u \, dv = uv - \int v \, du$

Set:

• $u = \frac{1}{x}$, so $du = -\frac{1}{x^2} dx$,
• $dv = \sin x \, dx$, so $v = -\cos x$.

Thus:

$\int_{1}^{\infty} \frac{\sin x}{x} \, dx = \left[-\frac{\cos x}{x}\right]_1^\infty + \int_{1}^{\infty} \frac{\cos x}{x^2} \, dx$

The first term, $-\frac{\cos x}{x}$, approaches zero as $x \to \infty$, since $\cos x$ oscillates between -1 and 1 and $x$ grows without bound. Therefore:

$\int_{1}^{\infty} \frac{\sin x}{x} \, dx = \int_{1}^{\infty} \frac{\cos x}{x^2} \, dx$

Step 2: Convergence of the Remaining Integral

To prove the convergence of

$\int_{1}^{\infty} \frac{\cos x}{x^2} \, dx,$

use the comparison theorem.

Bounding $|\cos x / x^2|$:

The absolute value of $\cos x$ satisfies $|\cos x| \leq 1$. Thus:

$\left| \frac{\cos x}{x^2} \right| \leq \frac{1}{x^2}.$

The integral $\int_{1}^{\infty} \frac{1}{x^2} dx$ is a p-integral with $p = 2 > 1$, so it converges. By the comparison theorem, the integral

$\int_{1}^{\infty} \frac{\cos x}{x^2} \, dx$

also converges.

Conclusion

Since the remaining integral converges, the original integral

$\int_{1}^{\infty} \frac{\sin x}{x} \, dx$

also converges.

Would you like a further breakdown of any specific step or additional examples related to improper integrals? Here are some follow-up questions to consider:

1. Can I provide more examples of integration by parts for improper integrals?
2. Would you like further details about the comparison theorem?
3. Do you need clarification on the behavior of oscillatory functions in improper integrals?
4. Should I explain convergence criteria for improper integrals in more depth?
5. Would you like a visual representation of how $\sin x / x$ behaves as $x \to \infty$?

Tip: Always analyze the absolute value of the integrand for convergence in oscillatory improper integrals.