Problem 18 - Applied Calculus

Question

Find each indefinite integral. \(\int\left(3 \sqrt{x}+\frac{1}{\sqrt{x}}\right) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \( 2x^{3/2} + 2x^{1/2} + C \).

Step 1: Rewrite the integrand in terms of exponents

To solve the integral \( \int\left(3 \sqrt{x}+\frac{1}{\sqrt{x}}\right) d x \), we begin by rewriting the square roots as exponents. Recall that \( \sqrt{x} = x^{1/2} \) and \( \frac{1}{\sqrt{x}} = x^{-1/2} \). Thus, the integral becomes: \[ \int \left( 3x^{1/2} + x^{-1/2} \right) dx \]

Step 2: Apply the power rule for integration

Integrate each term separately using the power rule of integration, which states that \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration and \( n eq -1 \). For \( 3x^{1/2} \): \[ \int 3x^{1/2} dx = 3 \cdot \frac{x^{1/2 + 1}}{1/2 + 1} = 3 \cdot \frac{x^{3/2}}{3/2} \] For \( x^{-1/2} \): \[ \int x^{-1/2} dx = \frac{x^{-1/2 + 1}}{-1/2 + 1} = \frac{x^{1/2}}{1/2} \]

Step 3: Simplify each integral term

First, simplify \( \int 3x^{1/2} dx \): \[ 3 \cdot \frac{x^{3/2}}{3/2} = 2x^{3/2} \]Next, simplify \( \int x^{-1/2} dx \): \[ \frac{x^{1/2}}{1/2} = 2x^{1/2} \]

Step 4: Combine the results with the constant of integration

Combine the simplified terms and include the constant of integration \( C \) to write the full indefinite integral: \[ 2x^{3/2} + 2x^{1/2} + C \]

Key Concepts

Power Rule of Integration

The Power Rule of Integration is a fundamental method for finding indefinite integrals, especially when working with polynomial functions. It provides a straightforward formula:

• If you need to integrate a term like \(x^n\), you increase the exponent by one and then divide by the new exponent, resulting in \(\frac{x^{n+1}}{n+1}\).
• Remember to always add the constant of integration, represented by \(C\), to signify the family of all possible antiderivatives.
• This method only works when \(n eq -1\), as dividing by zero is undefined. For \(n = -1\), the integral \(\int x^{-1} dx\) evaluates to \(\ln|x| + C\).

In the given exercise, we apply the Power Rule separately to each term after expressing them in exponent form. First, \(3x^{1/2}\) becomes \(2x^{3/2}\), and \(x^{-1/2}\) becomes \(2x^{1/2}\). This showcases how the Power Rule simplifies complex expressions by integrating each element independently.

Square Roots as Exponents

Understanding square roots as exponents is vital for applying integration techniques effectively. Here's how we translate square roots:

• The square root of \(x\) is expressed as \(x^{1/2}\).
• The reciprocal of a square root, \(\frac{1}{\sqrt{x}}\), is expressed as \(x^{-1/2}\).

This conversion is crucial because it allows us to use the Power Rule of Integration. By expressing terms in a polynomial form, we simplify the problem of finding the indefinite integral.For the specific problem, rewriting \(3 \sqrt{x} + \frac{1}{\sqrt{x}}\) as \(3x^{1/2} + x^{-1/2}\) allows us to use the Power Rule effectively. This step translates complex root expressions into a format amenable to polynomial integration.

Integration Constant

Whenever you perform indefinite integration, it's essential to remember the integration constant \(C\).

• This constant is crucial because indefinite integration represents a family of functions. The derivative of any of these functions results in the original integrand, differing only by a constant.
• Thus, \(C\) embodies all the possible vertical shifts along the y-axis that the antiderivative could undergo.

In the case of our exercise, adding \(C\) at the end of the integrated expression (\(2x^{3/2} + 2x^{1/2} + C\)) is essential. Without \(C\), the solution would only represent a single specific instance rather than the entire set of possible solutions. This makes it clear that there are infinitely many antiderivatives varying only by this constant.