Problem 34 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=\tan \left(\frac{x}{4}\right) $$

Step-by-Step Solution

Answer

The general antiderivative of \( f(x) = \tan\left(\frac{x}{4}\right) \) is \( -4 \ln\left|\cos\left(\frac{x}{4}\right)\right| + C \).

Step 1: Recognize the Antiderivative Formula

The function we need to find the antiderivative for is \( f(x) = \tan\left(\frac{x}{4}\right) \). For this function, it is useful to recognize that \( \tan(u) \) can be rewritten using the identity \( \tan(u) = \frac{\sin(u)}{\cos(u)} \). This suggests that integrating the tangent function might involve a logarithmic expression similar to \( \ln|\sec(u)| \). However, the integral \( \int \tan u \, du = -\ln|\cos u| + C \) is more straightforward to utilize in this situation.

Step 2: Identify the Substitution

Set \( u = \frac{x}{4} \), which makes the integral easier to solve. Therefore, we have \( du = \frac{1}{4} dx \) or \( dx = 4 du \). This substitution simplifies the function to \( \int \tan(u) \cdot 4 \, du \).

Step 3: Integrate with Respect to u

Substitute and integrate the expression \( 4 \int \tan(u) \, du \). The integration of \( \tan(u) \) is \( -\ln|\cos(u)| \). Thus, the integral becomes \( 4[-\ln|\cos(u)|] + C = -4 \ln|\cos(u)| + C \).

Step 4: Substitute Back for x

Replace \( u \) with \( \frac{x}{4} \) to express the antiderivative in terms of \( x \). Thus, the general antiderivative becomes \( -4 \ln\left|\cos\left(\frac{x}{4}\right)\right| + C \).

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is a fundamental trigonometric function. It is defined as the ratio of the sine and cosine functions:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

This relationship is crucial because it allows us to transform the tangent function into a form that is often easier to integrate. For instance, integrating \( \tan(u) \) can sometimes be directly tackled by transforming it into an expression involving \( \sin(u) \) and \( \cos(u) \).
The behavior of the tangent function is also interesting. It has vertical asymptotes where the cosine function is zero, leading to undefined values at odd multiples of \( \frac{\pi}{2} \). It is periodic with a period of \( \pi \), which means it repeats its shape every \( \pi \) units. Understanding these properties is essential when dealing with integrals and derivatives involving \( \tan(x) \) or its transformations.

Substitution Method

The substitution method is a powerful technique used in calculus to simplify complex functions for easier integration or differentiation. It involves replacing a part of the function with a single variable to make the operation more manageable. For instance, in the exercise provided, we perform the substitution \( u = \frac{x}{4} \).
By setting \( u = \frac{x}{4} \), we then calculate the differential \( du = \frac{1}{4} dx \) and express \( dx \) in terms of \( du \), which is \( dx = 4 du \). This transforms the integral of the given function into a more straightforward form:

• \( \int \tan\left(\frac{x}{4}\right) dx \rightarrow \int \tan(u) \cdot 4 du \)

Substitution streamlines the integration process, allowing you to work with a simpler basic function rather than complex expressions in terms of the original variable \( x \).
After integrating the simpler expression, we substitute back the original variable to achieve the final solution in its desired form.

Logarithmic Integration

Logarithmic integration is useful when you come across integrals that yield logarithmic functions as solutions. In particular, certain trigonometric functions like the tangent function often result in logarithmic expressions after integration. For the tangent function, the integral \( \int \tan(u) \, du \) evaluates to \( -\ln|\cos(u)| + C \), where \( C \) is the constant of integration.
The logarithmic expression arises from the identity transformation of \( \tan(u) \) into \( \frac{\sin(u)}{\cos(u)} \). During integration, focusing on \( \tan(u) \) as part of a recognizable derivative leads to the inclusion of a logarithm of the absolute value of the cosine function. It's an example of how logarithmic integration methods can simplify seemingly complex integrations by leaning on known formulas and transformations.
Thus, when completing the integration for \( \tan\left(\frac{x}{4}\right) \), after performing substitution, we integrate and then substitute back to reach \( -4\ln\left|\cos\left(\frac{x}{4}\right)\right| + C \), demonstrating how the original function elegantly transforms into a logarithmic result through calculation.