Problem 58 - Applied Calculus

Question

Find the indefinite integrals. $$ \int \sin (3 x) d x $$

Step-by-Step Solution

Answer

\(-\frac{1}{3} \cos(3x) + C\)

Step 1: Identify the Integral Formulation

The given integral is \( \int \sin(3x) \, dx \). This follows the form \( \int \sin(ax) \, dx \), where \( a = 3 \). We aim to find the antiderivative of \( \sin(3x) \).

Step 2: Recall the Antiderivative Rule for Sine Function

The antiderivative of \( \sin(ax) \) is \(-\frac{1}{a}\cos(ax) + C\), where \( C \) is the constant of integration. This rule helps us find the indefinite integral for our given function.

Step 3: Apply the Antiderivative Rule

Using the rule from Step 2, substitute \( a = 3 \) into \(-\frac{1}{a}\cos(ax) + C\) to get \(-\frac{1}{3} \cos(3x) + C \). This is the antiderivative of \( \sin(3x) \).

Step 4: Present the Solution

The indefinite integral of \( \sin(3x) \) is \(-\frac{1}{3} \cos(3x) + C\). Here, \( C \) represents the constant of integration.

Key Concepts

Antiderivative

Before diving into the solution of the given integral, it's essential to understand the concept of an antiderivative. An antiderivative of a function is, quite simply, a new function whose derivative yields the original function. In the context of this exercise, we have the sine function, \( \sin(3x) \). To find its antiderivative, we need to discover a function whose derivative is \( \sin(3x) \).

The notation for finding an antiderivative is known as integration, specifically indefinite integration when limits are not specified.

• An indefinite integral will produce an antiderivative of the function.
• The result will include a constant of integration, represented as \( C \).

Understanding the specific rules or formulas helps simplify the process of integration. The integral for a sine function follows a specific rule that dictates its integration process.

Sine Function

One of the most fundamental trigonometric functions, the sine function, regularly appears in calculus problems involving integration and differentiation. The sine function, noted as \( \sin(x) \), describes a wave-like pattern that oscillates between -1 and 1.

When tasked with integrating a sine function, there is a clear antiderivative rule to follow. For example, the antiderivative of the general sine function, \( \sin(ax) \), is \(-\frac{1}{a}\cos(ax) \).

• Here, \( a \) is a constant that influences the frequency of the oscillations.
• By finding the appropriate constant, one can obtain the correct antiderivative.

This rule provides the framework applied in our step-by-step solution of \( \sin(3x) \), where we've identified \( a = 3 \) and proceeded with the integration resulting in \(-\frac{1}{3}\cos(3x) \).

Constant of Integration

The constant of integration, often denoted as \( C \), is a critical component of indefinite integrals. When finding the antiderivative of a function, the solution is not unique; hence, \( C \) accounts for any additional constant that could exist.

Here's why it's important:

• Adding \( C \) ensures that all possible solutions are represented.
• Each different value for \( C \) represents a parallel shift of the function's graph.

In the case of indefinite integration of \( \sin(3x) \), the result is \(-\frac{1}{3}\cos(3x) + C \).This \( C \) is not tied to any specific value until further information, such as initial conditions, is applied. This approach reinforces the openness of solutions in calculus, accommodating a range of possible functions that share the same derivative.