Problem 2 - Calculus Single Variable

Question

Calculate the indefinite integral. $$ \int(3 \sin (x)-5 \cos (x)+1) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \(-3\cos(x) - 5\sin(x) + x + C\).

Step 1: Understand the Problem Statement

We need to find the indefinite integral of the function \( 3\sin(x) - 5\cos(x) + 1 \). This means we will find the antiderivative of the function with respect to \( x \).

Step 2: Break Down the Integral

The integral of a sum of functions is the sum of the integrals of the individual functions. Therefore, we break down our integral into three parts: \( \int 3\sin(x)\,dx \), \( -\int 5\cos(x)\,dx \), and \( \int 1\,dx \).

Step 3: Solve the First Integral

Find the integral \( \int 3\sin(x)\,dx \). The antiderivative of \( \sin(x) \) is \( -\cos(x) \). Therefore, \( \int 3\sin(x)\,dx = -3\cos(x) \).

Step 4: Solve the Second Integral

Solve \( -\int 5\cos(x)\,dx \). The antiderivative of \( \cos(x) \) is \( \sin(x) \). So, \( -\int 5\cos(x)\,dx = -5\sin(x) \).

Step 5: Solve the Third Integral

For the integral \( \int 1\,dx \), the antiderivative of a constant \( 1 \) is \( x \). Thus, \( \int 1\,dx = x \).

Step 6: Combine All Parts

Combine all the antiderivatives to get the complete antiderivative: \( -3\cos(x) - 5\sin(x) + x + C \), where \( C \) is the constant of integration.

Key Concepts

Antiderivative

Finding an antiderivative means identifying a function whose derivative equals the given function. In this exercise, the task is to find the indefinite integral, which essentially involves calculating the antiderivative of the function \(3\sin(x) - 5\cos(x) + 1\).

When solving for the antiderivative:

• Recognize that integration is the reverse operation of differentiation.
• Understand the basic rules of integration, such as the power rule and the constants multiplication rule.
• Separate terms in the integral to manage them individually.

For example, the antiderivative of \(\sin(x)\) is \(-\cos(x)\), and the antiderivative of \(\cos(x)\) is \(\sin(x)\). By finding these antiderivatives, you reverse the process of differentiation to obtain the original family of functions.

Trigonometric Functions

Trigonometric functions are crucial in calculus, often appearing in integrals and derivatives.

Some primary trigonometric functions include:

• \(\sin(x)\)
• \(\cos(x)\)
• \(\tan(x)\)

In our exercise, we're dealing specifically with \(\sin(x)\) and \(\cos(x)\). Integrating these functions involves knowing their antiderivatives:

• The antiderivative of \(\sin(x)\) is \(-\cos(x)\), because the derivative of \(-\cos(x)\) is \(\sin(x)\).
• The antiderivative of \(\cos(x)\) is \(\sin(x)\), since the derivative of \(\sin(x)\) is \(\cos(x)\).

Understanding these relationships allows you to handle more complex integrals involving trigonometric expressions with confidence.

Constant of Integration

When you find an indefinite integral, you need to add an arbitrary constant, known as the constant of integration, to the result. This is because differentiation removes constants, and integrating means retrieving all possible original functions.

Why do we need this constant?

• The process of finding an antiderivative does not determine an exact constant because a function and its shifted versions (by a constant) have the same derivative.
• Adding \(C\) ensures that your antiderivative represents all possible functions that could have been differentiated to get the original integrated function.

For example, in the solution \(-3\cos(x) - 5\sin(x) + x + C\), the \(C\) accounts for every possible vertical shift of the function on the graph, ensuring completeness of the result. This makes the mathematical representation of solutions entirely flexible until specific conditions (like initial conditions) are applied to specify \(C\'s\) exact value.