Problem 28 - Thomas Calculus

Question

In Exercises \(17-56,\) find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\frac{\sqrt{x}}{2}+\frac{2}{\sqrt{x}}\right) d x$$

Step-by-Step Solution

Answer

The antiderivative is \( \frac{x^{3/2}}{3} + 4 \sqrt{x} + C \).

Step 1: Simplify the Integrand

First, simplify the expression inside the integral. Rewrite the square roots in terms of exponents: \( \sqrt{x} = x^{1/2} \) and \( \frac{2}{\sqrt{x}} = 2x^{-1/2} \). Thus, the integral becomes: \[ \int \left( \frac{1}{2} x^{1/2} + 2 x^{-1/2} \right) dx. \]

Step 2: Apply the Power Rule for Integration

Apply the power rule for integration to each term. The power rule states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). For \( \frac{1}{2} x^{1/2} \), we have: \[ \int \frac{1}{2} x^{1/2} \, dx = \frac{1}{2} \cdot \frac{x^{3/2}}{3/2} = \frac{x^{3/2}}{3}. \] For \( 2 x^{-1/2} \), we have: \[ \int 2 x^{-1/2} \, dx = 2 \cdot \frac{x^{1/2}}{1/2} = 4 \sqrt{x}. \]

Step 3: Combine the Results

Combine the results of the integrations. Hence, the integral is: \[ \int \left( \frac{1}{2} x^{1/2} + 2 x^{-1/2} \right) dx = \frac{x^{3/2}}{3} + 4 \sqrt{x} + C, \] where \( C \) is the constant of integration.

Step 4: Verify by Differentiation

Differentiate the antiderivative to verify its correctness. The derivative of \( \frac{x^{3/2}}{3} \) is \( \frac{1}{2} x^{1/2} \), and the derivative of \( 4 \sqrt{x} \) is \( 2 x^{-1/2} \). Thus, the derivative is \( \frac{1}{2} x^{1/2} + 2 x^{-1/2} \), which matches the original integrand, confirming the correctness.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental tool when dealing with polynomials or terms that can be expressed as power functions. This rule simplifies the process of finding antiderivatives. Generally, it states that if you wish to integrate a function of the form \(x^n\), where \(neq -1\), the result will be \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
The constant \(C\) is crucial because integration is inverse differentiation, and differentiation of a constant returns zero.

To apply the power rule, ensure the integrand (the function being integrated) is expressed in terms of exponents. For example, \(\sqrt{x}\) is rewritten as \(x^{1/2}\), and \(\frac{2}{\sqrt{x}}\) as \(2x^{-1/2}\). This transformation aligns the terms with the power rule format.
As applied to the exercise, integrating \(\frac{1}{2}x^{1/2}\) gives \(\frac{x^{3/2}}{3}\) and integrating \(2x^{-1/2}\) gives \(4\sqrt{x}\). Both results are combined with a \(C\) to form the integral of the original function.

Antiderivative Verification

Verifying an antiderivative is vital to ensure that the integration process was carried out correctly. This involves differentiating the antiderivative you found and checking if it matches the original integrand.

For example, once the antiderivative \(\frac{x^{3/2}}{3} + 4 \sqrt{x} + C\) is found, its derivative should match the initial expression under the integral \(\frac{1}{2}x^{1/2} + 2x^{-1/2}\).
Apply basic differentiation rules to each term of your antiderivative:

• The derivative of \(\frac{x^{3/2}}{3}\) is \(\frac{1}{2}x^{1/2}\).
• The derivative of \(4\sqrt{x}\) is \(2x^{-1/2}\).

Summing these results gives you the original integrand, confirming the antiderivative's correctness. This step ensures the solution's accuracy and reinforces your understanding of integration and differentiation as inverse operations.

Simplifying Integrals

Simplifying an integral often involves rewriting terms into a more familiar or manageable form. Specifically, if the integrand involves roots, fractions, or complex expressions, consider converting them into power functions.

This strategy allows easier application of integration rules like the power rule. For instance, in the given exercise, the integrand \(\frac{\sqrt{x}}{2} + \frac{2}{\sqrt{x}}\) was simplified to \(\frac{1}{2}x^{1/2} + 2x^{-1/2}\).
This simplification reduces the cognitive load when applying integration techniques, helping to avoid mistakes.
Ultimately, simplifying before integrating can save time and make the process straightforward. It also lays a foundation where more complex integrals can be handled effectively using similar simplifications and appropriate integration rules.