Problem 73 - Calculus: Graphical, Numerical, Algebraic

Question

Multiple Choice \(\int \tan x d x=\) (A) \(\frac{\tan ^{2} x}{2}+C\) (B) \(\ln |\cot x|+C\) (C) \(\ln |\cos x|+C\) (D) \(-\ln |\cos x|+C\) (E) \(-\ln |\cot x|+C\)

Step-by-Step Solution

Answer

(D) \(-\ln |\cos x|+C\)

Step 1: Recall the Integral of Tangent Function

The integral of the tangent function, \( \tan x \), is a standard integral in calculus. It can be solved by remembering that the integration of \( \tan x \) with respect to \( x \) is the negative logarithm of the absolute value of cosine function. This is generally represented as \( -\ln |\cos x| \).

Step 2: Find the Correct Option

Now, check the given options and find the one which matches the calculated result. In this case, the correct result \( -\ln |\cos x| \) is given in option (D)

Step 3: Verify the Correct Option

While solving multiple choice questions, it can be a good practice to double check the calculations and make sure no information is missing or misunderstood. In our case, it can be done by taking the derivative of our answer and check if it matches with the original function. For option (D), the derivative of \( -\ln |\cos x| \) is \( \tan x \), the original function. Thus verifying our answer.

Key Concepts

Integration in Calculus

Integration is a fundamental concept in calculus that is often considered the reverse process of differentiation. When you integrate a function, you are essentially finding the area under the curve of that function. The result of an integration is called the integral and it can be thought of as the accumulation of a quantity over an interval.

For example, if you think about the speed of a car over time, the integration of the speed function will give you the total distance traveled. In our exercise, we were integrating the tangent function, which is a common trigonometric function. The process involved looking for a function that, when differentiated, would result in the tangent function, thereby 'undoing' the differentiation process.

There are various techniques used for integrating more complex functions such as integration by parts, substitution, and partial fractions. For trigonometric functions like the tangent, there is a standard result that makes the process straightforward, which is what we applied to solve the exercise given.

Natural Logarithm Properties

The natural logarithm, denoted as 'ln', is a logarithm to the base 'e', where 'e' is an irrational constant approximately equal to 2.71828. Logarithms have properties that make them particularly useful in solving calculus problems.

Some key properties of natural logarithms that are relevant to our exercise include:

• The logarithm of a product is the sum of the logarithms: \(\text{If } z = xy, \text{ then } \text{ln} \underline{\phantom{xxx}} z = \text{ln} \underline{\phantom{xxx}} x + \text{ln} \underline{\phantom{xxx}} y\).
• The logarithm of a quotient is the difference of the logarithms: \(\text{If } z = \frac{x}{y}, \text{ then } \text{ln} \underline{\phantom{xxx}} z = \text{ln}\underline{\phantom{xxx}}x - \text{ln}\underline{\phantom{xxx}}y\).
• The logarithm of a power is the exponent times the logarithm of the base: \(\text{If } z = x^y, \text{ then } \text{ln} \underline{\phantom{xxx}} z = y \underline{\phantom{xxx}} \text{ln} \underline{\phantom{xxx}} x\).

These properties allow us to manipulate logarithmic expressions conveniently when integrating and differentiating. In the context of the exercise, we used the natural logarithm of the absolute value of cosine to express the integral of the tangent function.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that contain trigonometric functions such as sine, cosine, tangent, and others. These types of integrals are common in calculus, and they often require special techniques or the use of trigonometric identities to solve.

Trigonometric functions have periodic properties and specific relationships with each other. For example, the tangent function is the quotient of sine over cosine, \( \tan x = \frac{\text{sin} x}{\text{cos} x} \).When we integrate the tangent function, we need to consider these relationships. The standard integral of tangent, as was necessary in our exercise, involves knowing that \( \frac{d}{dx}(-\text{ln} |\text{cos} x|) = \tan x \).

Understanding how trigonometric functions work and how they differentiate and integrate with respect to one another is a powerful tool in calculus. With practice, many integrals that include trigonometric functions can be solved by applying known identities and integrals.