Problem 12 - Calculus: Early Transcendental Functions

Question

Find the general antiderivative. $$\int(3 \cos x-\sin x) d x$$

Step-by-Step Solution

Answer

The antiderivative of \( \int(3 \cos x-\sin x) dx \) is \( 3\sin x -\cos x + c \), where \(c\) is the constant of integration.

Step 1: Separate the terms

First, separate the integral into two terms. \[ \int(3 \cos x-\sin x) dx = \int3 \cos x dx - \int \sin x dx \]

Step 2: Calculate the antiderivate of each term

The antiderivative of \(3 \cos x\) is \(3\sin x\) and the antiderivative of \(\sin x\) is \(-\cos x\). Therefore, we have \[ \int3 \cos x dx = 3\sin x + c_1 \] and \[ \int \sin x dx = -\cos x + c_2 \] where \(c_1\) and \(c_2\) are the constants of integration.

Step 3: Combine the results

Combine the two results to obtain the general antiderivative: \[ \int(3 \cos x-\sin x) dx = 3\sin x -\cos x + c \] where \(c = c_1 + c_2\) is the constant of integration.

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus, much like differentiation. It allows us to find a function's antiderivative, essentially reversing the differentiation process. There are various integration techniques available, each suitable for different types of functions. These methods help simplify complex problems and are crucial for solving integrals effectively. In the given exercise, we use the method of separating terms, known as term-by-term integration.

• First, you split a complex integral into smaller, simpler parts, integrating each term individually.
• This approach exploits linearity, allowing independent handling of each term, then recombining the results.

Another common technique is substitution, where we change variables to simplify the function before integrating. Also, integration by parts, which is helpful with products of functions, may be used. Understanding these techniques is key for tackling various integration challenges with ease.

Trigonometric Integrals

Trigonometric integrals involve functions of sine, cosine, tangent, and other trigonometric functions. They are common in calculus and require familiarity with trigonometric identities and their antiderivatives. In the exercise given, integrating trigonometric functions begins with understanding their derivatives and antiderivatives:

• The derivative of \( an x\) is \(rac{1}{ an^2 x} + 1\).
• The derivative of \( an^2(x)\) is \( an x \cdot 1\).
• The integral of \( an x\) is \(- an(x) \cdot 1\)
• So, by integrating \( an x\) and \( an^2(x)\), we can solve other trigonomic problems

For \(3 \cos x\), its antiderivative \(3 \sin x\) uses the fact that the derivative of \(sin x\) is \(cos x\).
Similarly, the antiderivative of \(sin x\) is \(-cos x\), since \(-cos x\) differentiates back to \(sin x\). Understanding these relationships allows effective handling of integrals involving trigonometric functions.

Constants of Integration

In the problem of finding antiderivatives, constants of integration play a crucial role. When integrating, you're not just finding one antiderivative, but a family of functions. Each has the same derivative, differing by a constant. This constant is the "constant of integration."
Whenever you integrate, you must introduce an arbitrary constant, often denoted by \(C\) or in the solution \(C_1\) and \(C_2\). They account for all the possible vertical shifts of the antiderivative function:

• Every function has infinitely many antiderivatives, each differing by a constant.
• This constant captures the "shift," differing solutions for functions have based on initial conditions.

In the exercise, after combining terms, a single constant of integration \(C = C_1 + C_2\) represents the cumulative effect of the individual constants from each term. It ensures the solution reflects the general form of all possible solutions, not just a single one.