91Ó°ÊÓ

Integral calculus is a core component of mathematical analysis dealing with accumulation of quantities and areas under curves. When we refer to integrals, we often mean finding the total accumulation of a quantity over a certain interval. In our exercise, we're focused on the concept of definite integrals, which help us determine the value of the function in a specific interval, in this case from 1 to 4.
Integrals can be visualized as the sum of infinitesimally small quantities. In essence, the definite integral of a function between two points a and b, noted as \(\int_{a}^{b} f(x) \, dx\), represents the net area between the function and the x-axis for this interval. A positive area is above the x-axis, while a negative area is below it.
Here's what happens in our exercise:- The expression inside the integral, \(\frac{3}{\sqrt{x}} - \frac{6}{\sqrt{x}}\), first needs simplification.- This simplifies the calculation and makes it easier to apply integration techniques later on.

The power rule is a fundamental principle used to integrate functions of the form \(x^n\). It states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration for indefinite integrals. For definite integrals, this simplifies the function without requiring the constant.
When we encounter expressions involving powers, like \(x^{-1/2}\) from our exercise, the power rule becomes particularly handy. In the problem, after simplifying, we rewrite the expression \(\frac{-3}{\sqrt{x}}\) as \(-3x^{-1/2}\). This step prepares the function for the application of the power rule:

• Increase the power by 1, making \(-1/2 + 1 = 1/2\).
• Divide by the new power \(1/2\) and multiply by the coefficient \(-3\), yielding \(-6x^{1/2}\).

By using this power rule, we solve the integral easily without complications.

Simplifying expressions is a crucial step in solving integrals efficiently. It allows us to transform complex functions into a form more suitable for integration. In our exercise, simplifying the expression inside the integral reduced \(\frac{3}{\sqrt{x}} - \frac{6}{\sqrt{x}}\) to \(\frac{-3}{\sqrt{x}}\), making the integral much more manageable to evaluate.
Here are the key steps we took during simplification:

• Identify and factor out common terms or expressions.
• Rewrite complex fractions using exponents; for example, converting \(\frac{1}{\sqrt{x}}\) to \(x^{-1/2}\).

This simplification is essential, not only for aiding in applying the power rule but also for ensuring a smooth and less error-prone calculation in further steps. In practical scenarios, simplification reduces the likelihood of mistakes in algebraic manipulations and leads to a clearer, more concise solution path.