Problem 32 - Applied Calculus

Question

Find each indefinite integral. \(\int \frac{z^{2}+1}{z} d z\)

Step-by-Step Solution

Answer

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

Step 1: Simplify the Integrand

The expression inside the integral is \(\frac{z^2 + 1}{z}\). Simplify this by dividing each term by \(z\). This results in \(z + \frac{1}{z}\). Thus, the integral becomes \(\int (z + \frac{1}{z}) \, dz\).

Step 2: Integrate Term by Term

Now, separate the integral into two parts: \(\int z \, dz + \int \frac{1}{z} \, dz\). These can be integrated separately.

Step 3: Integrate the First Term

The integral of \(z\) with respect to \(z\) is \(\frac{z^2}{2} + C_1\).

Step 4: Integrate the Second Term

The integral of \(\frac{1}{z}\) is \(\ln|z| + C_2\).

Step 5: Combine the Results

Combine the two results from Steps 3 and 4. The constants \(C_1\) and \(C_2\) can be combined into a single constant \(C\). Thus, the overall indefinite integral is \(\frac{z^2}{2} + \ln|z| + C\).

Key Concepts

Integrand Simplification

When tackling indefinite integrals, one key step is often simplifying the integrand, which is the expression inside the integral. Simplifying makes the integration process more manageable and sometimes reveals a straightforward path to a solution.

In our exercise, we're given the integrand \(\frac{z^2+1}{z}\). By dividing each term of the numerator by the denominator \(z\), we simplify the integrand to \(z + \frac{1}{z}\). This transformation is crucial because it separates the expression into more digestible parts. Simplification can involve:

• Breaking down complex fractions
• Factoring expressions
• Expanding powers (when necessary)

Once simplified, the integrand becomes easier to work with, paving the way for effective integration techniques. This approach showcases the power of algebraic manipulation in calculus.

Integration Techniques

With the integrand simplified, the next step is applying the right integration techniques. The integral is now \(\int (z + \frac{1}{z}) \, dz\). Integrating each term separately simplifies the process!

• For \(\int z \, dz\), use the power rule which states: The integral of \(z^n\) is \(\frac{z^{n+1}}{n+1}\).
• For \(\int \frac{1}{z} \, dz\), use the natural logarithm rule: The integral of \(\frac{1}{z}\) is \(\ln|z|\).

These techniques rely on the application of fundamental integration rules.
Given the simplicity of the terms, you avoid complex integration methods like substitution or integration by parts in this instance. Instead, you rely on straightforward and efficient rules that swiftly bring the solution into view.

Antiderivatives

The result of an indefinite integral is known as an antiderivative. It represents a function whose derivative gives back the original integrand.

Considering our example, we find that the antiderivative of \(z\) is \(\frac{z^2}{2}\), derived using the power rule, while the antiderivative of \(\frac{1}{z}\) is \(\ln|z|\), based on logarithmic integration.

However, indefinite integrals always include a constant of integration, \(C\), because a constant, when differentiated, turns to zero. This constant allows the representation of a family of functions, all of which are valid solutions. The final antiderivative in this exercise is \(\frac{z^2}{2} + \ln|z| + C\), which represents this family, accounting for all potential vertical shifts of the resulting curve.