Problem 24 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

In Exercises 21 through 30 , evaluate the indicated definite integral. \(\int_{1}^{9} \frac{x^{2}+\sqrt{x}-5}{x} d x\)

Step-by-Step Solution

Answer

44 - 5 \ln 9

Step 1: Simplify the Integrand

Simplify the integrand \ \( \frac{x^2 + \sqrt{x} - 5}{x} \) \ by separating it into individual terms: \ \( \frac{x^2}{x} + \frac{\sqrt{x}}{x} - \frac{5}{x} \). This simplifies to \ \( x + x^{-1/2} - 5x^{-1} \).

Step 2: Set Up the Integral

Set up the definite integral with the simplified integrand: \ \( \int_{1}^{9} \left( x + x^{-1/2} - 5x^{-1} \right) dx \).

Step 3: Integrate Each Term

Integrate each term separately: \ \( \int_{1}^{9} x \, dx \), \ \( \int_{1}^{9} x^{-1/2} \, dx \), and \ \( -5 \int_{1}^{9} x^{-1} \, dx \).

Step 4: Calculate the Antiderivatives

Calculate the antiderivatives for each term: \[ \int x \, dx = \frac{x^2}{2} \] \[ \int x^{-1/2} \, dx = 2x^{1/2} \] \[ \int x^{-1} \, dx = \ln|x| \].

Step 5: Evaluate the Definite Integrals

Evaluate the definite integral for each term using the upper and lower bounds: \ \( \left[ \frac{x^2}{2} \right]_{1}^{9} \), \ \( \left[ 2x^{1/2} \right]_{1}^{9} \), and \ \( -5 \left[ \ln|x| \right]_{1}^{9} \).

Step 6: Substitute the Bounds

Substitute the bounds into each antiderivative: \ \( \left[ \frac{9^2}{2} \right] - \left[ \frac{1^2}{2} \right] = \frac{81}{2} - \frac{1}{2} = 40 \), \ \( \left[ 2 \sqrt{9} \right] - \left[ 2 \sqrt{1} \right] = 6 - 2 = 4 \), \ \( -5 \left( \ln 9 - \ln 1 \right) = -5 \ln 9 \).

Step 7: Combine the Results

Combine the results of the evaluated integrals: \ \( 40 + 4 - 5 \ln 9 \). Thus, the value of the definite integral is \ \( 44 - 5 \ln 9 \).

Key Concepts

Integral Calculus

Integral calculus is a vital branch of mathematics that focuses on integration — a process of finding the area under a curve. In essence, integration is the reverse operation of differentiation. Integrals are classified into definite and indefinite integrals.

A definite integral, such as \(\int_{1}^{9} \frac{x^{2}+\sqrt{x}-5}{x} d x\), calculates the total accumulated value between given bounds (here, from 1 to 9). These numerical boundaries provide the shaded area directly.

On the contrary, an indefinite integral lacks specific boundaries and gives a general form of the anti-derivative (a function whose derivative is the integrand itself), denoted by the symbol \(\int\). These fundamental concepts help to analyze rates of change and cumulative quantities, forming a backbone for various scientific and engineering fields.

To better understand these integrals, break them into simpler parts, handle each one separately, and then combine the results.

Antiderivatives

The antiderivate, also known as the indefinite integral, is the function from which a given function is the derivative. For example, if you start from \(\frac{d}{dx}F(x) = f(x)\), then \(\int f(x) dx = F(x) + C\), where \(|C|\) is the constant of integration.

In the given exercise, we break down the integrand and find the antiderivative for each segment:

• For \(\int x dx\), the antiderivative is \(\frac{x^2}{2}\).
• For \(\int x^{-1/2} dx\), the antiderivative is \(\frac{2x^{1/2}}{1/2} = 2x^{1/2}\).
• For \(\int x^{-1} dx\), the antiderivative is \(\ln|x|\).

Each small piece of the integrand becomes a simpler term to handle. Understanding these antiderivatives ensures we get back to the original function during differentiation.

Antidifferentiation is foundational for solving problems involving area, accumulated change over time, and many other real-world applications.

Integration Techniques

Several techniques simplify the computation of integrals and handle complex functions efficiently. The primary integration techniques include:

• **Substitution**: This transforms the integrand into a simpler form by changing variables.
• **Integration by Parts**: Originating from the product rule for differentiation, this technique splits the integrand into products of functions.
• **Partial Fraction Decomposition**: Useful for rational functions, this breaks the integrand into simpler fractions before integrating.

For the given integral, simplifying the integrand was crucial. We divided \(\frac{x^{2}+\sqrt{x}-5}{x}\) into simpler terms:

• \(\frac{x^2}{x} = x\)
• \(\frac{\sqrt{x}}{x} = x^{-1/2}\)
• \(\frac{-5}{x} = -5x^{-1}\)

This step-by-step simplification makes each term integrable directly. Afterward, integrating each piece separately and combining the results allowed us to determine the definitive integral value.

Utilizing these techniques appropriately is critical for handling more complex integral problems effectively.