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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!

Evaluate the definite integral: ∫(5/x^2) dx from x = 1 to x = 4.

Learn how to evaluate the definite integral ∫(5/x^2) dx from 1 to 4 with step-by-step calculations. This problem covers the power rule for integration and the application of the Fundamental Theorem of Calculus. Ideal for students in Grades 11-12 and college-level calculus.

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