Problem 64 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \sin \frac{x}{3} d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral to Solve

We need to find the indefinite integral \( \int \sin \frac{x}{3} \, dx \). This is an integral involving a trigonometric function.

Step 2: Use Substitution Method

To integrate \( \sin(\frac{x}{3}) \), we can use the substitution method. Let \( u = \frac{x}{3} \), which implies \( du = \frac{1}{3} dx \), or equivalently \( dx = 3 \, du \).

Step 3: Rewrite the Integral

Substitute \( u \) and \( du \) into the integral: \[ \int \sin \frac{x}{3} \, dx = \int \sin(u) \, 3 \, du = 3 \int \sin(u) \, du\]

Step 4: Integrate with Respect to u

The integral \( \int \sin(u) \, du \) is a standard integral, which equals \( -\cos(u) + C \), where \( C \) is the constant of integration. Therefore, \[ 3 \int \sin(u) \, du = 3(-\cos(u)) + C = -3 \cos(u) + C \].

Step 5: Substitute Back to x

Replace \( u \) with \( \frac{x}{3} \) to get the answer in terms of \( x \): \[ -3 \cos(u) + C = -3 \cos\left(\frac{x}{3}\right) + C \].

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in calculus and appear frequently in integration problems. In this exercise, the focus is on the sine function, denoted as \( \sin \). These functions are periodic and have well-defined integrals.

• The sine function, \( \sin(x) \), oscillates between -1 and 1 and has a period of \( 2\pi \).
• Integrating trigonometric functions can often lead to other trigonometric functions, which is why it's important to understand their properties and identities.

Recognizing when a trigonometric function is part of an integral is the first step in performing the integration process.
In this exercise, the integrand contains the sine function with a modified argument, \( \sin\left(\frac{x}{3}\right) \). This kind of expression is a common type and requires tailored substitution techniques for integration.

Substitution Method

The substitution method, or \( u \)-substitution, is a technique used to simplify the process of integrating functions. It's particularly useful when dealing with trigonometric integrals where the argument of the trig function is not just \( x \). Here’s how it works in this example:

• We choose \( u \) to represent a part of the integral's argument, \( u = \frac{x}{3} \). This simplification reduces complexity because it allows us to rewrite the integral in terms of \( u \).
• The derivative of \( u \) with respect to \( x \) is \( du = \frac{1}{3} dx \), which we rearrange to get \( dx = 3 \, du \).

By substituting \( u \) and \( du \) back into the integral, the expression becomes \( 3\int \sin(u) \, du \), which is easier to integrate.
Substitution method essentially transforms an otherwise difficult integral into one that involves more familiar, standard forms. It simplifies the original problem by detaching it from the original variable, allowing us to focus solely on the simpler form and providing a clear path to the solution.

Constant of Integration

In indefinite integrals, the solution always involves a constant of integration denoted as \( C \). This constant is added because the process of integration is the reverse of differentiation, which means many different functions could have the same derivative.

• The constant \( C \) represents all the possible vertical shifts of the antiderivative on a graph. Since differentiation removes any constant, integrating can only determine these constants up to their possible variations.
• It’s crucial to include \( C \) in your final answer for indefinite integrals to correctly represent all families of solutions.

In this problem, after performing the integration using \( u \)-substitution and integrating \( \sin(u) \) to get \( -\cos(u) \), we arrive at the solution containing \( -3 \cos\left(\frac{x}{3}\right) + C \).
Without the constant, the solution would only describe a subset of all possible antiderivatives, lacking completeness.