Problem 33 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=\tan \left(\frac{x}{3}\right) $$

Step-by-Step Solution

Answer

\( F(x) = -3 \ln \left| \cos \left( \frac{x}{3} \right) \right| + C \)

Step 1: Understand the Problem

We need to find the general antiderivative of the function \( f(x) = \tan \left( \frac{x}{3} \right) \). This means we are seeking a function \( F(x) \) such that \( F'(x) = \tan \left( \frac{x}{3} \right) \).

Step 2: Recall an Integration Rule

We know that the general antiderivative of \( \tan(u) \) is \( -\ln |\cos(u)| + C \), where \( C \) is the constant of integration. We will use this rule while performing substitution.

Step 3: Substitution

Let \( u = \frac{x}{3} \). Then \( \frac{du}{dx} = \frac{1}{3} \), which implies \( dx = 3 \, du \). Substituting these into the integral, we have:\[ \int \tan \left( \frac{x}{3} \right) dx = \int \tan(u) \, 3 \, du = 3 \int \tan(u) \, du \]

Step 4: Integrate with Respect to \( u \)

Integrating \( \tan(u) \) gives:\[ 3 \int \tan(u) du = 3 [-\ln |\cos(u)| + C] = -3 \ln |\cos(u)| + 3C \]

Step 5: Substitute Back the Original Variable

Re-substitute \( u = \frac{x}{3} \) back into the expression:\[ -3 \ln \left| \cos \left( \frac{x}{3} \right) \right| + 3C \]Thus, the general antiderivative \( F(x) \) of \( f(x) \) is:\[ F(x) = -3 \ln \left| \cos \left( \frac{x}{3} \right) \right| + C \] where \( C = 3C \) is an arbitrary constant.

Key Concepts

Trigonometric Integration

Trigonometric integration is a technique used to find the integral of trigonometric functions. When dealing with these functions, it's crucial to remember their derivatives and antiderivatives. For example, we recall that the derivative of \( \tan(u) \) is \( \sec^2(u) \) and its antiderivative is \( -\ln |\cos(u)| + C \), where \( C \) is the constant of integration. Trigonometric integrals often arise in various fields of science and engineering, and mastering them is essential.
Using these known integral properties can greatly simplify the process of finding antiderivatives, provided we correctly identify the trigonometric function and its integral counterpart. A powerful aspect of trigonometric integration is the ability to transform an initially complicated problem into a much simpler one by detecting patterns and adjusting accordingly through substitution or other methods.

Substitution Method

The substitution method is a handy technique used to simplify integration problems by changing variables. It's similar to the chain rule in differentiation, but in reverse. This method involves setting the variable that makes the problem complex into a simpler form. In the given exercise, we used substitution by defining \( u = \frac{x}{3} \). This transform helped convert the function \( \tan\left(\frac{x}{3}\right) \) into \( \tan(u) \), making the integration much more straightforward.

• Calculate the derivative of the substitution variable, in this case, \( \frac{du}{dx} = \frac{1}{3} \).
• From here, solve for \( dx \) in terms of \( du \) to convert the entire integral into the new variable.

This substitution led to the integral \( 3 \int \tan(u) \, du \). The factor of \( 3 \) arises from solving for \( dx \). Finally, once the integration is carried out in terms of \( u \), it's important to substitute back the original variable to represent the final antiderivative correctly. This step completes the process and switches the context back to the variable initially involved in the problem.

Integration Rules

Integration rules are fundamental tools necessary for evaluating integrals efficiently. Memorizing basic integration rules can assist greatly in solving problems quickly and effectively. The integral of a function is essentially the reverse process of differentiation, thus recognizing derivative forms is crucial in identifying their antiderivatives.
For instance, in this exercise, the integration rule utilized was the antiderivative of \( \tan(u) \), which is \( -\ln |\cos(u)| + C \). Recognizing this relationship is what allows the process to continue smoothly after substitution has simplified the function.

• Understand common trigonometric identities and how they relate to their integrals.
• Apply these rules consistently once substitutions or transformations have been completed.

Remember, consistent practice of these foundational integration rules and concepts can vastly improve your ability to tackle complex integrals with confidence and accuracy. These rules are like stepping stones, guiding you through the problem until you achieve the desired solution.