Let's analyze the integrals in the image.

a)

$\int_1^5 \frac{1}{\sqrt{x^4 - 1}} \, dx$ To determine whether this integral converges or diverges, we need to check if there are any points where the integrand becomes undefined or leads to infinite values.

• The denominator $\sqrt{x^4 - 1}$ becomes 0 when $x = 1$ or $x = -1$, but since the lower limit of integration is $1$, we need to analyze the behavior near $x = 1$.
• As $x \to 1^+$, the denominator approaches $0$, which suggests a potential singularity. We can investigate this using a limit or approximation.

b)

$\int_1^\infty \frac{1}{\sqrt{x + x^3}} \, dx$ We will evaluate the asymptotic behavior as $x \to \infty$.

• For large $x$, the term $x^3$ dominates, so $\sqrt{x + x^3} \approx x^{3/2}$.
• Hence, $\frac{1}{\sqrt{x + x^3}} \sim \frac{1}{x^{3/2}}$.
• We compare this with the integral $\int_1^\infty \frac{1}{x^{3/2}} \, dx$, which converges, so this integral also converges.

c)

$\int_1^\infty \frac{\sqrt{2x + 1} \ln x}{x^4} \, dx$ For large $x$, the factor $\sqrt{2x + 1} \sim \sqrt{2x}$, so the integrand behaves like: $\frac{\sqrt{2x} \ln x}{x^4} \sim \frac{\ln x}{x^{7/2}}$ We compare this with the integral $\int_1^\infty \frac{\ln x}{x^{7/2}} \, dx$, which converges, so this integral also converges.

d)

$\int_1^\infty e^{-1 - x^2} \, dx$ The exponential function $e^{-x^2}$ decays very rapidly for large $x$. As $x \to \infty$, the integrand approaches $0$ exponentially fast. Therefore, the integral converges.

All the integrals in the problem converge. If you have any questions or need further clarification on any step, feel free to ask!

Further Exploration:

1. What are the techniques for handling integrals with singularities at the limits of integration?
2. How do we determine the convergence of improper integrals involving infinite limits?
3. How can we apply the comparison test to determine the convergence of integrals?
4. What is the role of exponential functions in determining the convergence of integrals?
5. How do we deal with integrals that involve both polynomial and logarithmic terms?

Tip: For integrals with infinite limits, often the key to convergence is the behavior of the integrand for large values of the variable.