Problem 29 - Calculus: Early Transcendentals

Question

Evaluate the integral. \( \displaystyle \int^4_1 \frac{2 + x^2}{\sqrt{x}} \,dx \)

Step-by-Step Solution

Answer

The integral evaluates to 16.4.

Step 1: Rewrite the Integrand

Start by rewriting the integrand \( \frac{2 + x^2}{\sqrt{x}} \) in terms of powers of \( x \). We can separate this into two separate fractions: \( \frac{2}{\sqrt{x}} + \frac{x^2}{\sqrt{x}} \). Simplify these fractions to get: \( 2x^{-1/2} + x^{3/2} \).

Step 2: Set Up the Integral

Write out the integral with the simplified integrand:\[\int_1^4 \left( 2x^{-1/2} + x^{3/2} \right) \, dx\]

Step 3: Integrate Each Term

Integrate each term of the expression separately.1. For \( \int 2x^{-1/2} \, dx \), use the power rule: \[ \text{Result: } 2 \cdot \frac{x^{1/2}}{1/2} = 4x^{1/2} \]2. For \( \int x^{3/2} \, dx \), also use the power rule: \[ \text{Result: } \frac{x^{5/2}}{5/2} = \frac{2}{5}x^{5/2} \]

Step 4: Evaluate the Indefinite Integral

Combine the results from Step 3 to form the indefinite integral:\[ 4x^{1/2} + \frac{2}{5}x^{5/2} + C \]

Step 5: Evaluate the Definite Integral

Substitute the limits of integration 1 and 4 into the indefinite integral:\[ \left[ 4x^{1/2} + \frac{2}{5}x^{5/2} \right]_1^4 \]Calculate:- For \( x = 4 \): \[ 4 \cdot 4^{1/2} + \frac{2}{5} \cdot 4^{5/2} = 4 \cdot 2 + \frac{2}{5} \cdot 32 = 8 + \frac{64}{5} = 8 + 12.8 = 20.8 \]- For \( x = 1 \): \[ 4 \cdot 1^{1/2} + \frac{2}{5} \cdot 1^{5/2} = 4 \cdot 1 + \frac{2}{5} \cdot 1 = 4 + 0.4 = 4.4 \]Finally, compute:\[ 20.8 - 4.4 = 16.4 \]

Step 6: Conclusion

The value of the integral \( \int_1^4 \frac{2 + x^2}{\sqrt{x}} \, dx \) is 16.4.

Key Concepts

Integration Techniques

Integrating functions involves various techniques that help simplify the process of finding an integral. Understanding these techniques is crucial for evaluating complex integrals effectively. In this example, the integral \[ \int^4_1 \frac{2 + x^2}{\sqrt{x}} \,dx \]requires dealing with both algebraic manipulation and the application of integration rules. To handle such integrals, consider the following approaches:

• **Algebraic Manipulation:** Start by breaking down complex expressions into simpler parts that are easier to integrate. This often involves separating terms and simplifying them.
• **Substitution:** Although not used in this particular problem, substitution can be a helpful technique when you need to replace variables to simplify the integrand.
• **Integration by Parts:** This technique applies when you have a product of functions, often necessary for more complex expressions.

By selecting the appropriate technique based on the characteristics of the integrand, you can streamline the integration process and handle a variety of functions.

Power Rule

The power rule is a fundamental tool in integration that helps solve integrals involving power functions. It simplifies the integration process, allowing you to convert the function into a simpler form. The power rule states that for any constant power \( n \), the integral of \( x^n \) is given by:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C,\] where \( C \) is the constant of integration. This rule is particularly effective for rational powers of \( x \), as seen in this exercise.

• For the term \( 2x^{-1/2} \), applying the power rule gives:\[ 2 \times \frac{x^{1/2}}{1/2} = 4x^{1/2}. \]
• Similarly, for the term \( x^{3/2} \), you get:\[ \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2}. \]

Understanding and using the power rule allows for quick evaluation of terms, making it a staple in solving integrals involving polynomial or rational expressions. Always remember to adjust the constant in the expression properly.

Rewriting Integrals

Rewriting integrals is a helpful strategy that focuses on transforming a given integrand into a form that is easier to integrate. This technique often involves algebraic manipulation to change the structure of the expression.For the integral \[ \int^4_1 \frac{2 + x^2}{\sqrt{x}} \,dx, \]the first step is rewriting the integrand. We separate and simplify the expression:

• Divide each term by \( \sqrt{x} \):\[ \frac{2}{\sqrt{x}} + \frac{x^2}{\sqrt{x}}. \]
• Rewrite using powers of \( x \) : the integrand becomes:\[ 2x^{-1/2} + x^{3/2}. \]

This step transforms the original complicated expression into a more manageable form, setting the stage for applying integration techniques like the power rule. Rewriting integrals is a critical skill that helps simplify and break down even the most intricate functions, easing the path to finding solutions.