Problem 57 - Applied Calculus

Question

Find the indefinite integrals. $$ \int(3 \cos x-7 \sin x) d x $$

Step-by-Step Solution

Answer

\( \int (3 \cos x - 7 \sin x) \, dx = 3 \sin x + 7 \cos x + C \).

Step 1: Separate the Integral

To find the indefinite integral of \( \int (3 \cos x - 7 \sin x) \, dx \), first split it into two separate integrals: \( \int 3 \cos x \, dx \) and \( -\int 7 \sin x \, dx \).

Step 2: Factor Out Constants

For the first integral \( \int 3 \cos x \, dx \), factor out the constant 3: \( 3 \int \cos x \, dx \). For the second integral \(-\int 7 \sin x \, dx \), factor out the -7: \(-7 \int \sin x \, dx \).

Step 3: Integrate Cosine Function

The integral of \( \cos x \) is \( \sin x \). Thus, \( \int 3 \cos x \, dx = 3 \sin x \).

Step 4: Integrate Sine Function

The integral of \( \sin x \) is \(-\cos x \). Thus, \(-\int 7 \sin x \, dx = 7 \cos x \).

Step 5: Combine Results and Add Constant

Combine the results from Step 3 and Step 4: \( 3 \sin x + 7 \cos x + C \), where \( C \) is the constant of integration.

Key Concepts

Integral Calculus

Integral calculus is a fundamental branch of calculus concerned with the concept of integration. In integral calculus, we focus on finding functions given their rates of change, which is essentially reversing differentiation. This method is essential for calculating areas under curves and solving problems involving accumulation.

An **indefinite integral**, also known as an antiderivative, refers to finding a function whose derivative equals the integrand (the function inside the integral). In the case of our exercise, we are looking for a function whose derivative with respect to x is \(3 \cos x - 7 \sin x\).

Indefinite integrals are written with the integral symbol \(\int\), and since they originate functions, they include a constant of integration, denoted as \(C\), because differentiating with any constant results in zero. Thus, integrating involves adding this constant to account for any vertical shifts in the graph of the antiderivative.

• Calculates areas and accumulations
• Reverse process of differentiation
• Includes a constant of integration \(C\)

Trigonometric Integrals

Trigonometric integrals are specific types of integrals involving trigonometric functions such as sine, cosine, tangent, etc. Solving these integrals usually requires knowledge of the basic antiderivatives of trigonometric functions.

For instance, the antiderivative of \(\cos x\) is \(\sin x\), while the antiderivative of \(\sin x\) is \(-\cos x\). In the given problem, both trigonometric functions are present:

• \(\int 3 \cos x \, dx\) gives \(3 \sin x\)
• \(-\int 7 \sin x \, dx\) gives \(7 \cos x\)

These trigonometric identities are pivotal for finding the indefinite integral.

Using these basic results simplifies integration, allowing us to efficiently solve problems involving trigonometric functions.

Integration Techniques

Various integration techniques help us solve more complex integrals. This exercise involves one of the simplest techniques: basic antiderivatives and constant factoring.

Here's a brief overview of the steps used in integrating our function:

• **Separation:** Separate the integrand into easily manageable parts. For example, \(\int(3 \cos x - 7 \sin x) dx\) can be broken into \(\int 3 \cos x \, dx\) and \(-\int 7 \sin x \, dx\).
• **Factoring Constants:** Extract constants out of the integral. Hence, \(3\) and \(-7\) were factored out, simplifying our task to finding \(\int \cos x \, dx\) and \(\int \sin x \, dx\).
• **Basic Antiderivatives:** Use known results from calculus; for example, \(\int \cos x \, dx = \sin x\) and \(\int \sin x \, dx = -\cos x\).

Breaking complex problems into simpler tasks make integration smooth and straightforward. Familiarity with these techniques builds confidence in handling varied integral problems.