Problem 43 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The integral is \(2w^{1.5} + C\).

Step 1: Rewrite the Integral

The integrand \(3 \sqrt{w}\) can be rewritten for easier integration. Recall that the square root \(\sqrt{w}\) is the same as \(w^{0.5}\). So, the integral becomes \(\int 3 w^{0.5} \, dw\).

Step 2: Apply the Power Rule for Integration

To integrate \(3 w^{0.5}\), use the power rule for integration, which states \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\). Here, \(n = 0.5\).

Step 3: Perform the Integration

Using the power rule, integrate \(3 w^{0.5}\):\[\int 3 w^{0.5} \, dw = 3 \cdot \frac{w^{0.5 + 1}}{0.5 + 1} + C = 3 \cdot \frac{w^{1.5}}{1.5} + C\]

Step 4: Simplify the Expression

Simplify the expression \(3 \cdot \frac{w^{1.5}}{1.5}\):\[3 \times \frac{1}{1.5} = 2\]Thus, the integral becomes\[2w^{1.5} + C\].

Step 5: Write the Final Answer

The indefinite integral of \(3 \sqrt{w}\) is \(2w^{1.5} + C\), where \(C\) is the constant of integration.

Key Concepts

Power Rule for Integration

The power rule for integration is a crucial technique in calculus. It allows us to find the integral of functions that can be expressed as a power of a variable. The rule states that for any real number \(n eq -1\), the integral \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \(C\) is the constant of integration. This rule is derived from the reverse process of differentiation.

• To apply the power rule, you add one to the exponent.
• Then divide by the new exponent.
• Don't forget the constant of integration, \(C\).

In the context of our exercise, the function \(3 \sqrt{w}\) was rewritten as \(3w^{0.5}\), making it easier to apply the power rule. After rewriting, you simply integrate using the power rule: increase the exponent by one \((0.5 + 1 = 1.5)\) and then divide by this new exponent, resulting in an integral of the form \(3 \cdot \frac{w^{1.5}}{1.5} + C\). This simplifies further to \(2w^{1.5} + C\), representing the indefinite integral.

Square Root Function

The square root function is a common function that appears in calculus problems. It's important to recognize that a square root can be rewritten using exponents. Specifically, given \(\sqrt{w}\), this can be expressed as \(w^{0.5}\). Writing the function in this way is useful because it allows us to apply techniques such as the power rule for integration.

• The exponent \(0.5\) signifies the square root, which can be thought of as a power of one-half.
• Rewriting functions as exponents simplifies the process of integration.
• This method can be applied to various roots, not just the square root.

Recognizing and converting the square root into an exponent form is a critical step in solving integration problems involving roots. In our problem, writing \(3 \sqrt{w}\) as \(3w^{0.5}\) simplifies it into a format where the power rule for integration can be easily applied.

Integration Techniques

To solve integrals effectively, understanding and choosing the right integration technique is key. Here are some commonly used techniques:

• Power Rule: Useful for polynomials and functions that are a power of \(x\).
• Substitution: Needed when functions are composed or more complex, like involving another function inside them.
• Integration by Parts: Typically applied when dealing with products of functions, such as a polynomial and an exponential.

With these techniques, problems that initially seem complex may become more manageable. In our exercise, the power rule was the most direct approach since the problem involved a simple power function after rewriting the square root.
By understanding the nature of the function involved in the integral \(3 \sqrt{w}\), we could see that rewriting the square root function allowed us to directly apply the power rule, simplifying our problem-solving process significantly.