The given integral is:

$\int_{1}^{+\infty} \frac{x + 1}{\sqrt{x^4 - x}} \, dx$

To evaluate this integral, let's analyze it step by step:

Step 1: Examine the convergence of the integral

Before solving, we need to ensure the integral converges, particularly since it has an improper upper limit at $+\infty$.

As $x \to \infty$:

The numerator $x + 1$ grows linearly, while the denominator $\sqrt{x^4 - x} \sim x^2$. Therefore, the integrand $\frac{x+1}{\sqrt{x^4 - x}} \sim \frac{x}{x^2} = \frac{1}{x}$. The improper integral $\int \frac{1}{x} \, dx$ diverges logarithmically at infinity.

Conclusion

The integral diverges because the integrand does not decay quickly enough as $x \to \infty$. Thus:

$\int_{1}^{+\infty} \frac{x + 1}{\sqrt{x^4 - x}} \, dx \text{ diverges.}$

Would you like further details or explanations about improper integrals or convergence criteria? Let me know!

Here are 5 questions related to this analysis:

1. What are the criteria for determining whether an improper integral converges?
2. How do we compare the growth rates of functions in improper integrals?
3. What happens if we evaluate the integral only over a finite interval (e.g., $1$ to a large finite value)?
4. How would a substitution like $u = x^2$ change the problem?
5. Can this problem be analyzed differently using numerical methods or graphing?

Tip:

When dealing with improper integrals, always verify convergence before attempting to evaluate!