Problem 18 - Applied Calculus

Question

Find each integral by using the integral table on the inside back cover. $$ \int \frac{\sqrt{4+z^{2}}}{z} d z $$

Step-by-Step Solution

Answer

The integral is \( \sqrt{4+z^2} - 2 \ln|z+\sqrt{4+z^2}| + C \).

Step 1: Recognize the Integral Form

Examine the integral \( \int \frac{\sqrt{4 + z^2}}{z} \, dz \) and identify a form that matches any standard expressions in the integral table available.

Step 2: Match with Integral Table Entry

Find the entry in the integral table that corresponds to \( \int \frac{\sqrt{a^2 + z^2}}{z} \, dz \), where \( a^2 = 4 \), which means \( a = 2 \). The table indicates that the antiderivative is a form involving logarithms and inverse trigonometric functions.

Step 3: Apply the Matching Formula

Using the table's formula for \( \int \frac{\sqrt{a^2 + z^2}}{z} \, dz \), apply it with \(a = 2\) to get:\[ \int \frac{\sqrt{4 + z^2}}{z} \, dz = \sqrt{4 + z^2} - 2 \ln|z + \sqrt{4 + z^2}| + C,\]where \(C\) is the constant of integration.

Key Concepts

Integral Table

When encountering a complex integral, like \( \int \frac{\sqrt{4+z^{2}}}{z} \, dz \), students often rely on an integral table to simplify their calculations. An integral table is a comprehensive collection of functions and their integrals. It acts as a shortcut for finding antiderivatives of complex or less straightforward integrals.
This can save time and reduce errors, particularly for integrals that do not lend themselves to easy manipulation or substitution.

• Integral tables list various forms, such as \( \int \frac{\sqrt{a^2 + x^2}}{x} \, dx \), and their solved results.
• They provide standard results which are especially useful in exams or time-constrained settings.
• Identifying the correct form helps in applying the right formula from the table to obtain the solution immediately.

By matching your integral with one in the table, you can find a direct path to the solution. In our exercise, we found a match for \( \int \frac{\sqrt{4+z^{2}}}{z} \, dz \), simplifying our task thanks to the integral table.

Antiderivative

The antiderivative, also known as the indefinite integral, of a function represents a key concept in integral calculus, allowing us to find a function, \( F(x) \), such that its derivative is equal to the given function, \( f(x) \). In simple terms, the antiderivative is the reverse process of differentiation.
In the context of the exercise, finding the antiderivative of \( \frac{\sqrt{4+z^{2}}}{z} \) involved using known integral forms.

• The antiderivative addresses the question of accumulating a quantity or reversing the process of differentiation.
• In solving \( \int \frac{\sqrt{4+z^{2}}}{z} \, dz \), we identify a pattern and apply the corresponding antiderivative from the integral table.
• The result \( \sqrt{4 + z^2} - 2 \ln|z + \sqrt{4 + z^2}| + C \) includes the constant \( C \), reflecting the infinite family of solutions typical of antiderivatives.

The pursuit of antiderivatives is a fundamental aspect of integral calculus, vital in solving real-world problems involving accumulation.

Inverse Trigonometric Functions

Inverse trigonometric functions arise naturally in integral calculus, especially in scenarios involving squared terms or identities. These functions help in expressing angles as values of classic trigonometric functions.
The solution to \( \int \frac{\sqrt{4+z^{2}}}{z} \, dz \) interestingly involves a logarithmic term that holds similarities with the expressions involving inverse trigonometric functions.
The use of inverse trigonometric functions in antiderivatives is common when integrals involve forms like \( a^2 + x^2 \).

• Such functions like \( \arcsin \), \( \arccos \), and \( \arctan \) provide a meaningful way to represent angles based on given values.
• Though not directly evident in the final result of this exercise, these functions often appear in related integral forms.
• They provide a versatile toolset for expressing solutions where trigonometric identities are involved.

Understanding inverse trigonometric functions enhances your ability to solve complex integrals, as seen in many integral table entries and calculus problems.