Problem 24 - Applied Calculus

Question

Find each indefinite integral. \(\int\left(\frac{1}{z^{2}}+\frac{1}{\sqrt[3]{z}}\right) d z\)

Step-by-Step Solution

Answer

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

Step 1: Simplify the integrand

First, rewrite the integrand in a more manageable form for integration. The expression \( \frac{1}{z^{2}} \) can be rewritten as \( z^{-2} \), and \( \frac{1}{\sqrt[3]{z}} \) can be rewritten as \( z^{-1/3} \). Thus, the integrand becomes \( z^{-2} + z^{-1/3} \).

Step 2: Integrate the first term

To find \( \int z^{-2} \, dz \), use the power rule for integration, which states that \( \int z^n \, dz = \frac{z^{n+1}}{n+1} + C \), provided \( n eq -1 \). Applying this, \( \int z^{-2} \, dz = \frac{z^{-1}}{-1} = -z^{-1} = -\frac{1}{z} \).

Step 3: Integrate the second term

Now, find \( \int z^{-1/3} \, dz \). Using the power rule again, \( n = -1/3 \), so \( \int z^{-1/3} \, dz = \frac{z^{2/3}}{2/3} = \frac{3}{2} z^{2/3} \).

Step 4: Combine the results and add the constant of integration

Combine the results from the previous steps and write the final integral with the constant of integration, \( C \). This gives \( \int \left( z^{-2} + z^{-1/3} \right) \, dz = -\frac{1}{z} + \frac{3}{2} z^{2/3} + C \).

Key Concepts

Power Rule for Integration

The power rule for integration is a straightforward technique that is pivotal in solving many integral problems. Especially when dealing with polynomials or expressions that can be rewritten in polynomial form, it becomes particularly useful. The rule states:

• \( \int z^n \, dz = \frac{z^{n+1}}{n+1} + C \) for \( n eq -1 \)

This rule allows us to integrate terms with real-number exponents by increasing the exponent by one and dividing by this new exponent.
It's important to ensure that the exponent \( n \) is not equal to -1 because this would lead to a division by zero. In such a case, a different approach would be required, for example, through the natural logarithm integration technique,

• \( \int \frac{1}{z} \, dz = \ln |z| + C \)

In our problem, the terms \( z^{-2} \) and \( z^{-1/3} \) were rewritten to clearly apply the power rule. This makes integrating straightforward, allowing the transformation of expressions into antiderivatives.

Integration Techniques

Integration techniques are methods used to find the antiderivative or integral of a function. Each technique fits different types of integrals, and choosing the right technique is crucial for solving calculus problems effectively.
Some common techniques include integration by parts, substitution, and the power rule.

• Integration by Parts: Useful when integrands are products of functions, typically in the form of \( \int u \, dv = uv - \int v \, du \).
• Substitution: Great for expressions where a change of variables simplifies the integral, often used for composite functions.
• The Power Rule: As covered, is perfect for polynomials or functions that can be expressed in terms of powers of \( z \).

Selecting the appropriate method can simplify the problem-solving process. For our problem, by first simplifying the terms \( \frac{1}{z^2} \) and \( \frac{1}{\sqrt[3]{z}} \) into \( z^{-2} \) and \( z^{-1/3} \), the power rule is the best fit to effectively compute the indefinite integral.

Calculus Problem Solving

Solving calculus problems often involves breaking down the problem into simpler, manageable parts. This involves understanding the functions, recognizing patterns, and applying suitable mathematical rules and techniques.
For integration problems like the one given, it's essential to:

• Simplify expressions where possible, as we did by rewriting the integrand terms in their exponent forms.
• Choose suitable integration techniques, such as the power rule, to straightforwardly solve parts of the integral.
• Combine results carefully, ensuring each term is correctly integrated and the constant of integration \( C \) is added.

This structured approach helps navigate the sometimes complex landscape of calculus problems. By taking one step at a time, solutions come together more naturally, ensuring clarity and reducing the likelihood of errors.