Problem 28 - Thomas Calculus in SI Units

Question

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 \( \frac{1}{3}x^{3/2} + 4x^{1/2} + C \).

Step 1: Rewrite the Integrand

The original integrand is \( \frac{\sqrt{x}}{2} + \frac{2}{\sqrt{x}} \). Rewrite it using exponents for easier integration. Recall that \( \sqrt{x} = x^{1/2} \) and \( \frac{1}{\sqrt{x}} = x^{-1/2} \), so the integrand becomes \( \frac{1}{2}x^{1/2} + 2x^{-1/2} \).

Step 2: Apply the Power Rule for Integration

Use the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Apply it to each term separately: \( \int \frac{1}{2}x^{1/2} \, dx = \frac{1}{2} \times \frac{x^{3/2}}{3/2} = \frac{1}{3}x^{3/2} \). Similarly, for the second term, \( \int 2x^{-1/2} \, dx = 2 \times \frac{x^{1/2}}{1/2} = 4x^{1/2} \).

Step 3: Combine the Integrals and Add Constant

Combine the results from the previous step and remember to add the constant of integration, \( C \), since we're finding an indefinite integral. The antiderivative is \( \frac{1}{3}x^{3/2} + 4x^{1/2} + C \).

Step 4: Verify by Differentiation

Differentiate the result to check if it matches the original integrand: Compute \( \frac{d}{dx}\left(\frac{1}{3}x^{3/2} + 4x^{1/2} + C\right) \). This gives \( x^{1/2} \times \frac{1}{2} \cdot \frac{3}{3} + 4 \times x^{-1/2} \times \frac{1}{2} = \frac{x^{1/2}}{2} + \frac{2}{x^{1/2}} \), which matches the original expression.

Key Concepts

Power Rule for Integration

The power rule for integration is a handy tool for evaluating integrals, especially indefinite ones. It is similar to the power rule for differentiation but works in reverse. This rule helps us find antiderivatives more easily. When you have an integral of the form \( \int x^n \, dx \), where \( n eq -1 \), you can apply the power rule:

• Increase the exponent by 1, making it \( n+1 \).
• Divide by this new exponent, \( n+1 \).

This method gives you the antiderivative \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
Consider the term \( \frac{1}{2}x^{1/2} \). Applying the power rule:

• Raise \( x^{1/2} \) to \( x^{3/2} \).
• Divide by \( 3/2 \), giving \( \frac{1}{3}x^{3/2} \).

Now, apply the same rule to the term \( 2x^{-1/2} \):

• Raise \( x^{-1/2} \) to \( x^{1/2} \).
• Divide by \( 1/2 \), resulting in \( 4x^{1/2} \).

These steps help simplify the process of finding antiderivatives.

Antiderivative

An antiderivative, also known as the indefinite integral, is a function whose derivative gives the original function you started with. When you integrate, you're essentially reversing the process of differentiation. This means finding the antiderivative restores the original function from its rate of change.
In simpler terms, if you start with \( f'(x) \), the derivative of some function \( f(x) \), integrating \( f'(x) \) returns you to \( f(x) \). But with indefinite integrals, remember to add a constant of integration \( C \). This constant represents any potential vertical shift in your function's curve, as differentiation causes this information to be lost.
In the example from the problem, earlier steps showed that \( \frac{1}{3}x^{3/2} + 4x^{1/2} + C \) is the antiderivative. It means the derivative of this expression would take us back to the original function, \( \frac{\sqrt{x}}{2} + \frac{2}{\sqrt{x}} \). So, the "most general" antiderivative means capturing every possible version of the function with potential constants.

Verification by Differentiation

Verification by differentiation is a crucial step in ensuring the correctness of your integral solution. It involves taking the derivative of the antiderivative you found and checking whether it matches the original integrand. This process acts as a confirmation that the integral was solved correctly.
For example, let's consider the expression \( \frac{1}{3}x^{3/2} + 4x^{1/2} + C \), which we've identified as the antiderivative.

• Differentiate \( \frac{1}{3}x^{3/2} \), which results in \( \frac{3}{3}x^{1/2} \) or simply \( x^{1/2} \).
• Differentiate \( 4x^{1/2} \), which yields \( 2x^{-1/2} \).

Adding these derivative results gives the original integrand \( \frac{x^{1/2}}{2} + \frac{2}{x^{1/2}} \). Since the differentiation led us back to the initial integrand, it confirms that the antiderivative calculation was correct.
This verification method is always recommended after finding an antiderivative, as it ensures both accuracy and understanding of the integral problem.