Problem 60 - Applied Calculus

Question

Find the indefinite integrals. $$\int 3 \sqrt{w} d w$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Integral

The given integral is \( \int 3 \sqrt{w} \, dw \). The goal is to find an antiderivative of the function \( 3 \sqrt{w} \). We can express the square root as a power.

Step 2: Re-Express the Integrand

We know that \( \sqrt{w} = w^{1/2} \). Thus, the integrand \( 3 \sqrt{w} \) can be rewritten as \( 3w^{1/2} \). We can now express the integral as \( \int 3w^{1/2} \, dw \).

Step 3: Apply the Power Rule for Integration

To integrate \( 3w^{1/2} \), apply the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) where \( n eq -1 \). Here, \( n = \frac{1}{2} \).

Step 4: Integrate the Expression

First, add 1 to the exponent: \( \frac{1}{2} + 1 = \frac{3}{2} \). Then, divide the variable raised to this new power by the new exponent: \( 3 \times \frac{w^{3/2}}{3/2} = 3 \times \frac{2}{3} w^{3/2} = 2w^{3/2} \).

Step 5: Add the Constant of Integration

Don't forget to include the constant of integration \( C \) at the end of the process, as this is an indefinite integral. Thus, the final answer is \( 2w^{3/2} + C \).

Key Concepts

Antiderivative

In the world of calculus, an antiderivative is a function that reverses the action of a derivative. Let's break this down.

• When you differentiate a function, you are finding its rate of change.
• Conversely, when you find an antiderivative, you are attempting to recover the original function from its rate of change.

The indefinite integral, often denoted by the integral sign \( \int \), is used to represent the family of all possible antiderivatives of a given function. In practical terms, finding an antiderivative is like asking "what function could have been differentiated to get this result?" This is crucial in solving various problems in calculus, like integrating \( 3 \sqrt{w} \) to determine its antiderivative.

Power Rule for Integration

The power rule for integration is a fundamental technique used for finding indefinite integrals.

• This rule helps us integrate functions in the form \( x^n \), where \( n \) is not equal to -1, by systematically considering the power to which \( x \) is raised.
• The rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

In simple terms, we add 1 to the exponent and then divide by this new exponent to get our result.
This formula makes integrating powers of \( x \) straightforward and is fundamental to calculus. For example, integrating \( 3w^{1/2} \) by this rule, we first convert the square root into a power, \( w^{1/2} \), then perform the integration methodically to achieve the correct result.

Constant of Integration

When dealing with indefinite integrals, the constant of integration, represented as \( C \), is of utmost importance.
Because differentiation wipes out any constant added to a function, indefinite integrals could potentially represent an infinite number of functions.

• Including \( C \) ensures all potential original functions are accounted for in our solution.
• It represents that any of these constants could have been the starting point before differentiation happened.

Thus, when solving indefinite integrals like \( \int 3 \sqrt{w} \, dw \), adding \( C \) at the end acknowledges all possible antiderivatives. Without it, we'd miss a critical aspect of calculus solutions.