Problem 10 - Applied Calculus

Question

Find each indefinite integral. \(\int \frac{d z}{\sqrt[3]{z}}\)

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{3}{2} z^{\frac{2}{3}} + C \).

Step 1: Rewrite the Integral

The integral is given as \( \int \frac{d z}{\sqrt[3]{z}} \). First, let's rewrite the integrand using exponent notation. Recall that \( \sqrt[3]{z} \) is equivalent to \( z^{\frac{1}{3}} \). Thus, \( \frac{1}{\sqrt[3]{z}} = z^{-\frac{1}{3}} \). Now the integral is \( \int z^{-\frac{1}{3}} \, d z \).

Step 2: Apply the Power Rule for Integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \). Applying this to our integral, where \( n = -\frac{1}{3} \), we have: \( \int z^{-\frac{1}{3}} \, dz = \frac{z^{-\frac{1}{3} + 1}}{-\frac{1}{3} + 1} + C \).

Step 3: Simplify the Expression

Simplify the expression found in Step 2. The exponent \( -\frac{1}{3} + 1 \) simplifies to \( \frac{2}{3} \). The denominator \( -\frac{1}{3} + 1 \) also simplifies to \( \frac{2}{3} \). So the integral becomes: \( \frac{z^{\frac{2}{3}}}{\frac{2}{3}} + C \).

Step 4: Rationalize the Result

To further simplify, take the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \), and multiply it with \( z^{\frac{2}{3}} \). Therefore, the integral \( \int z^{-\frac{1}{3}} \, dz \) simplifies to \( \frac{3}{2} z^{\frac{2}{3}} + C \).

Key Concepts

Power Rule for Integration

When you encounter an integral, the power rule for integration is a useful technique to simplify and solve it. This rule applies to functions of the form \( x^n \) and is given by the formula:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

where \( n e -1 \) and \( C \) is the constant of integration.
The power rule essentially reverses the process of differentiation. For example, if you differentiate \( \frac{x^{n+1}}{n+1} \), you get back \( x^n \).

To apply the power rule in integration:

• Identify the power \( n \) of the variable.
• Add 1 to the power.
• Divide by the new power.
• Never forget the constant of integration \( C \).

Whether it's a simple polynomial or a complex expression transformed into a suitable form, the power rule makes integration straightforward and approachable.

Exponent Notation

Exponent notation is a concise way of expressing powers of a number. For instance, \( \sqrt[3]{z} \) is the cube root of \( z \) and can be written as \( z^{1/3} \). Understanding exponent notation is key to solving calculus problems efficiently.

Let's consider the expression \( \frac{1}{\sqrt[3]{z}} \). By using exponent notation, you express the denominator as \( z^{1/3} \), and the quotient becomes \( z^{-1/3} \).

• Any root expression can be transformed into an exponent form. For example, \( \sqrt{x} \) becomes \( x^{1/2} \).
• Fractional exponents indicate roots, while negative exponents represent reciprocals.

By rewriting expressions in this manner, it makes use of the power rule easier and often clearer. In calculus, manipulating powers using exponent notation turns complex expressions into more manageable forms.

Calculus Problems

Solving calculus problems frequently involves using integration and differentiation. These processes are fundamentally about understanding how functions change and accumulate values.

Let's delve into solving a typical problem: the indefinite integral. The goal is to find a function whose derivative is the given function. By integrating, you capture all possible functions through the constant of integration \( C \).

Consider the indefinite integral \( \int \frac{dz}{\sqrt[3]{z}} \). To approach this:

• First, rewrite the expression using exponent notation, \( z^{-1/3} \).
• Next, apply the power rule for integration.
• Simplify the resulting expression.
• Write the final answer including the constant \( C \).

Each step helps to build a deeper understanding of calculus concepts. With practice, solving such problems can enhance your ability to see the underlying structure in mathematical expressions, paving the way for tackling more complex challenges.