Problem 27 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The general antiderivative is \(-3 \cos\left(\frac{x}{3}\right) + 3 \sin\left(\frac{x}{3}\right) + C\).

Step 1: Identify the Problem

We need to find the antiderivative (indefinite integral) of the given function \( f(x) = \sin\left(\frac{x}{3}\right) + \cos\left(\frac{x}{3}\right) \). This means we are looking for a function \( F(x) \) such that \( F'(x) = f(x) \).

Step 2: Break Down the Function

The function is composed of two separate trigonometric terms: \( \sin\left(\frac{x}{3}\right) \) and \( \cos\left(\frac{x}{3}\right) \). We will integrate each term separately and then combine the results.

Step 3: Use the Substitution Method

Let's substitute \( u = \frac{x}{3} \), then \( \frac{du}{dx} = \frac{1}{3} \) or \( dx = 3 \, du \). This substitution will simplify the integration process for both terms.

Step 4: Integrate the Sine Term

The integral of \( \sin(u) \) with respect to \( u \) is \( -\cos(u) \). Applying the substitution, we have:\[\int \sin\left(\frac{x}{3}\right) \, dx = -3 \cos\left(\frac{x}{3}\right)\]

Step 5: Integrate the Cosine Term

The integral of \( \cos(u) \) with respect to \( u \) is \( \sin(u) \). Applying the substitution, we have:\[\int \cos\left(\frac{x}{3}\right) \, dx = 3 \sin\left(\frac{x}{3}\right)\]

Step 6: Combine the Results

Combining the integrated terms from Steps 4 and 5, the general antiderivative of the function is:\[F(x) = -3 \cos\left(\frac{x}{3}\right) + 3 \sin\left(\frac{x}{3}\right) + C\]where \( C \) is the constant of integration.

Key Concepts

Indefinite Integral

An indefinite integral is essentially the reverse operation of differentiation. When we talk about finding the indefinite integral of a function, we aim to find a general formula that describes antiderivatives of the function. This means we're looking for all possible functions which, if you differentiate them, would give back the original function we started with. In this exercise, we're focusing on the indefinite integral of the function:\[f(x)=\sin \left(\frac{x}{3}\right)+\cos \left(\frac{x}{3}\right)\]Finding the indefinite integral means determining the function \(F(x)\) such that \(F'(x) = f(x)\). The result will include a constant of integration \(C\), reflecting that there are infinitely many antiderivatives for a given function. In our solution, this means looking for \(F(x)\) which is the general form:\[F(x)= -3 \cos \left(\frac{x}{3}\right) + 3 \sin \left(\frac{x}{3}\right) + C\]

Trigonometric Functions

Trigonometric functions like sine and cosine are periodic functions that are fundamental in many areas of mathematics, including calculus. When you take the indefinite integral of a trigonometric function, you find another trigonometric function.

• The integral of \(\sin(x)\) is \(-\cos(x)\).
• The integral of \(\cos(x)\) is \(\sin(x)\).

In the given exercise, the function \(\sin \left(\frac{x}{3}\right)+\cos \left(\frac{x}{3}\right)\) involves both \(\sin\) and \(\cos\) functions. Understanding these intefrals helps calculate their indefinite integrals effectively:

• For \(\sin\left(\frac{x}{3}\right)\), you move towards \(-\cos\left(\frac{x}{3}\right)\), adjusting by constants from the substitution (covered next).
• For \(\cos\left(\frac{x}{3}\right)\), you arrive at \(\sin\left(\frac{x}{3}\right)\), again adjusting similarly with the substitution method.

Substitution Method

The substitution method is a crucial technique in calculus to simplify integration of complex functions. This approach often makes the integration process much more manageable by transforming it into a form that's easier to handle. In this problem, we encounter a situation where direct integration is complicated, and substitution is ideal.

Setting the Substitution

For the function \(f(x)\), substituting \(u = \frac{x}{3}\) simplifies our integrations. The differential \(dx\) becomes a function of \(du\), specifically, \(dx = 3 \, du\).

Applying to the Function

When applying this substitution:

• The \(\sin\left(\frac{x}{3}\right)\) term becomes \(-3 \cos(u)\) as \(\int \sin(u) \, du = -\cos(u)\).
• The \(\cos\left(\frac{x}{3}\right)\) term becomes \(3 \sin(u)\) as \(\int \cos(u) \, du = \sin(u)\).

Reversing the Substitution

After finding the antiderivatives in terms of \(u\), revert back to \(x\) by replacing \(u\) with \(\frac{x}{3}\), giving us the integrated function. Combing these results and adding the constant of integration \(C\), we acquire the general antiderivative:\[F(x) = -3 \cos\left(\frac{x}{3}\right) + 3 \sin\left(\frac{x}{3}\right) + C\]