The given expression is a definite integral:

$\int_1^4 \frac{5}{x^2} \, dx$

Let's solve this step by step.

1. The integrand is $\frac{5}{x^2} = 5x^{-2}$.

2. The antiderivative of $5x^{-2}$ is:

$\int 5x^{-2} \, dx = 5 \cdot \left( \frac{x^{-1}}{-1} \right) = -\frac{5}{x}$

3. Now, we apply the limits of integration, 1 and 4:

$-\frac{5}{x} \Bigg|_1^4 = -\frac{5}{4} + \frac{5}{1}$

4. Simplifying:

$-\frac{5}{4} + 5 = 5 - \frac{5}{4} = \frac{20}{4} - \frac{5}{4} = \frac{15}{4}$

Thus, the value of the definite integral is:

$\frac{15}{4}$

Would you like any more details, or do you have further questions?

Here are five related questions for practice:

1. What is the integral of $\frac{1}{x^2}$ without the constant multiplier?
2. How would you apply integration to solve a real-world problem involving rates of change?
3. What is the result of integrating $\frac{1}{x^3}$ over the same limits?
4. How would you solve $\int_1^4 \frac{5}{x^3} dx$?
5. What happens if we change the limits of integration from $[1,4]$ to $[2,5]$?

Tip: Always simplify the integrand before integrating—it often makes the computation easier!