Problem 63 - Essential Calculus

Question

Show that the two formulas are equivalent. $$ \begin{array}{l} \int \tan x d x=-\ln |\cos x|+C \\ \int \tan x d x=\ln |\sec x|+C \end{array} $$

Step-by-Step Solution

Answer

The two integral formulas for \(\tan x\) are shown to be equivalent because \(\sec x = \frac{1}{\cos x}\), and with the application of relevant logarithm properties, we arrive at the same formula.

Step 1: Understand the properties of trigonometric functions

It is known from basic trigonometric identities that \(\cos x = \frac{1}{\sec x}\).The absolute value function in logarithms can be ignored when proving equivalence as the individual identities are equivalent.

Step 2: Substitute the trigonometric identity into the second formula

Substitute \(\sec x= \frac{1}{\cos x}\) into the second formula, thus \(\ln |\sec x|= \ln |\frac{1}{\cos x}|\).

Step 3: Use the logarithmic property

Using the logarithmic property of \(\ln(a/b) = \ln(a) - \ln(b)\) and \(\ln(1/a)=-\ln(a)\), we can convert \(\ln |\frac{1}{\cos x}|\) into \(- \ln |\cos x|\).

Step 4: Comparison

Compare the formula derived with the first formula. They are both \(-\ln |\cos x|+C\), meaning they are equivalent.

Key Concepts

Trigonometric Identities

When working with trigonometric integrals, understanding trigonometric identities is crucial. These identities allow us to transform one expression into another that can be more squarely related to what we want to solve.
For this particular exercise, the identity \( \cos x = \frac{1}{\sec x} \) plays an essential role. Essentially, this identity reveals that the cosine of an angle is the reciprocal of its secant.
These relationships are found in standard trigonometry, and knowing them can simplify complex integrals.

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \cos^2 x + \sin^2 x = 1 \)

Using these identities can make calculus problems, like finding the equivalence of integrals, easier to solve. Next time you face a trig integral, think of the identities and how they might be applied to simplify your work.

Logarithmic Properties

Logarithmic properties are useful tools in calculus for simplifying expressions involving logarithms. One important property is that the logarithm of a quotient is the difference of the logarithms.
This is represented as \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \). When using this property, you can break down complex logarithmic expressions into simpler parts, making them easier to manage.
Another feature of logarithms that helps in this context is \( \ln\left(\frac{1}{a}\right) = -\ln(a) \). This means that taking the logarithm of a reciprocal leads to a negative logarithm.

• Change of Base: \( \log_b M = \frac{\log_k M}{\log_k b} \)
• Product to Sum: \( \ln(ab) = \ln(a) + \ln(b) \)
• Power Rule: \( \ln(a^n) = n \cdot \ln(a) \)

Applying these properties correctly lets you simplify and compare integrals, as demonstrated in the solution by transforming \( \ln | \sec x | \) into \(-\ln | \cos x | \). Such properties are invaluable for calculus students.

Integral Equivalence

The concept of integral equivalence is fundamentally about demonstrating that two seemingly different integral expressions can yield the same result. In this exercise, we have two integrals that are initially presented in different forms but are proven equivalent.
The task involves converting \( \int \tan x \, dx \) into two expressions: \(-\ln |\cos x| + C\) and \(\ln |\sec x| + C\). By applying trigonometric identities and logarithmic properties, we find that these expressions are, in fact, equal.
This idea of equivalence is central in calculus, as it allows flexibility in how integral results can be expressed.

• Understanding the relationship between different functions and identities can open multiple pathways for solving integrals.
• The ability to recognize equivalent forms is a skill that is developed with practice and familiarity with mathematical properties.

With practice, recognizing the equivalence between different integral representations will become a powerful tool in your mathematical toolkit.