Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \int(\sin \theta-\cos \theta) d \theta $$

Step-by-Step Solution

Answer

\(-\cos \theta + \sin \theta + C\)

Step 1: Break Down the Integral

We need to integrate the function \( \sin \theta - \cos \theta \). We can do this by integrating each term individually.

Step 2: Integrate Each Term

Integrate \( \sin \theta \) and \( - \cos \theta \) separately.1. The integral of \( \sin \theta \) is \( - \cos \theta \).2. The integral of \( - \cos \theta \) is \( - \sin \theta \).Thus, \[ \int (\sin \theta - \cos \theta) \, d\theta = \int \sin \theta \, d\theta - \int \cos \theta \, d\theta = - \cos \theta + \sin \theta. \]

Step 3: Add the Constants of Integration

As this is an indefinite integral, we add a constant of integration \( C \) to our result. Therefore, the result of the integral is \[ \boxed{ -\cos \theta + \sin \theta + C } \].

Key Concepts

Indefinite Integrals

When we talk about indefinite integrals, we're referring to a type of integral that doesn't have specified limits of integration. This means that rather than calculating the area under a curve for a specific interval, indefinite integrals provide a general form of an antiderivative. This general form includes a constant known as the constant of integration.

• This constant of integration, denoted by \( C \), accounts for the fact that any antiderivative could differ by a constant value in the differentiation process.
• The basic idea is to reverse the process of differentiation in achieving an integral.

Adding \( C \) helps in encompassing all possible antiderivatives of the function. It’s important to note that integrating a function reverses what differentiation does, returning us to the original function up to an additive constant. In our exercise, after integrating each term separately, we include \( C \) to formulate the general solution: \( -\cos \theta + \sin \theta + C \).

Trigonometric Functions

Trigonometric functions such as sine and cosine form the foundation of numerous calculations in mathematics and engineering. These functions are inherently periodic and manifest in wave-like patterns.

• Understanding how to integrate these functions is crucial because they frequently appear in various calculus problems.
• The integral of trivially differentiable trigonometric functions is often 'flipped' or reversed in sign and nature.

For instance:- The integral of \( \sin \theta \) is \( -\cos \theta \), reflecting the differentiation rule where the derivative of \( \cos \theta \) is \(-\sin \theta \).- Similarly, the integral of \( -\cos \theta \) is \( -\sin \theta \), reversing the derivative of \( \sin \theta \).Mastering these fundamental ideas about trigonometric integrals helps simplify problems where these functions appear.

Step-by-Step Solutions

Breaking down problems into manageable steps is a vital approach in mathematics that aids understanding and problem-solving. In the context of integrating functions, a step-by-step approach helps isolate each part of the problem. In our specific example:1. **Break Down the Integral:** Start by recognizing that the integral \( \int (\sin \theta - \cos \theta) \, d\theta \) can be split. Each term can be integrated separately which simplifies the process.2. **Integrate Each Term:** We address each function individually. - Integrating \( \sin \theta \) provides \( -\cos \theta \). - Integrating \( -\cos \theta \) yields \( -\sin \theta \).3. **Add the Constant of Integration:** After obtaining the antiderivatives, append the constant \( C \) to incorporate all possible solutions.This methodical breakdown not only achieves the solution but helps reinforce understanding by focusing on each component's role in the overall integral.