Problem 28 - Applied Calculus

Question

Find an antiderivative. $$f(x)=5 x-\sqrt{x}$$

Step-by-Step Solution

Answer

The antiderivative of \(f(x)=5x-\sqrt{x}\) is \(F(x)=\frac{5}{2}x^2-\frac{2}{3}x^{3/2}+C\).

Step 1: Identify the Function Types

The function \( f(x) = 5x - \sqrt{x} \) consists of two terms: a linear term, \(5x\), and a root term, \(\sqrt{x}\), which is \(x^{1/2}\). We will find the antiderivative of each term separately.

Step 2: Antiderivative of the Linear Term

The antiderivative of a linear term \( ax \) is \( \frac{1}{2}ax^2 + C \). For \(5x\), the antiderivative is calculated as follows:\[ \int 5x \, dx = \frac{5}{2}x^2 + C_1 \]

Step 3: Antiderivative of the Root Term

The antiderivative of \( x^{n} \) is \( \frac{1}{n+1}x^{n+1} + C \). For \( \sqrt{x} = x^{1/2} \), the antiderivative is:\[ \int x^{1/2} \, dx = \frac{1}{1/2 + 1}x^{1/2 + 1} + C_2 = \frac{2}{3}x^{3/2} + C_2 \]

Step 4: Combine the Antiderivatives

To find the antiderivative of the entire function, combine the results from Steps 2 and 3. The combined antiderivative is:\[ F(x) = \frac{5}{2}x^2 - \frac{2}{3}x^{3/2} + C \]Here, \( C = C_1 - C_2 \) is an arbitrary constant representing the sum of the constants from each term.

Key Concepts

Integration Techniques

Integration is the process of finding the antiderivative or the integral of a function. It's the reverse operation of differentiation. Various techniques can be applied to integrate different functions. By understanding these techniques, you can solve various integral problems more easily.

A crucial aspect of integration involves handling each part of a complex function individually. For example, the function given in the exercise is composed of a linear term and a root term. These terms have different derivatives, so they also require different integration approaches.

• Linear terms: Use simple integration rules like the power rule for integration to find the antiderivative of polynomials.
• Non-linear terms: Apply rules based on the form of the term, such as the root power in the exercise, which involves fractional or negative exponents.

By recognizing each part of a function, you can apply an appropriate strategy from your integration toolkit to conveniently solve the problem.

Linear Functions

Linear functions are among the simplest kinds of mathematical expressions. A linear function looks like \(ax + b\), where \(a\) and \(b\) are constants. They graph as straight lines, hence the name 'linear'. Understanding how to handle linear functions during integration can be incredibly useful because they frequently occur in algebra and calculus.

The antiderivative of a linear term \(ax\) follows an easy-to-remember formula:

• Integrate \(ax\) as: \(\frac{a}{2}x^2 + C\)

This rule is straightforward because the power of \(x\) increases by one, and you divide the coefficient by this new power. Applying this to the exercise, you can see how the term \(5x\) results in an antiderivative of \(\frac{5}{2}x^2\).

Linear function integration is foundational, allowing you to handle other more complicated problems with confidence.

Power Rule

One of the core rules of integration is the power rule. This rule greatly simplifies finding antiderivatives for polynomial functions and various other expressions. As long as the exponent isn't \(-1\), the power rule can be used.

The power rule for integration states:

• To integrate \(x^n\), you calculate \(\frac{1}{n+1}x^{n+1} + C\)

Here, \(n\) is the exponent and \(C\) is the constant of integration. Because the exercise included a root term \(\sqrt{x}\), which is \(x^{1/2}\), the power rule applies. By setting \(n = \frac{1}{2}\), you find the antiderivative to be \(\frac{2}{3}x^{3/2}\).

Mastering the power rule empowers you to break down and solve integrals swiftly and efficiently. It's one of the first integration tools any calculus student should have in their mathematical arsenal.