Problem 7 - Calculus with Concepts in Calculus

Question

Find all antiderivatives of the given function. $$ \sin x $$

Step-by-Step Solution

Answer

The antiderivatives of \( \sin x \) are \(-\cos x + C\).

Step 1: Understand the Problem

We need to find all antiderivatives (or indefinite integrals) of the given function, \( \sin x \). This means we have to determine a function \( F(x) \) such that its derivative, \( F'(x) \), equals \( \sin x \).

Step 2: Recall Basic Integration Formulas

One of the basic integration formulas is that the antiderivative of \( \sin x \) is \( -\cos x + C \), where \( C \) is the constant of integration. This is because the derivative of \( -\cos x \) is \( \sin x \).

Step 3: Write Down the General Antiderivative

Based on the basic integration formula, the general solution for the antiderivative of \( \sin x \) is \(-\cos x + C\). The constant \( C \) accounts for any constant term that can be added to the antiderivative since the derivative of a constant is zero.

Key Concepts

Integration Formulas

Integration formulas serve as the foundation for finding antiderivatives. Antiderivatives are functions whose derivative is the original function given in the problem. For example, if you have a function such as \( \sin x \), its antiderivative can be identified using these formulas.

• Basic Trigonometric Integrals: These include the integrals of basic trigonometric functions like \( \sin x \), \( \cos x \), and \( \tan x \). For instance, the antiderivative of \( \sin x \) is \( -\cos x + C \).
• Power Rule for Integration: This rule helps integrate functions of the form \( x^n \).

By referring to these formulas, you can more easily determine the function whose derivative results in the initial function you started with. It's beneficial to memorize these integral formulas, making it easier to tackle antiderivative problems efficiently. Each integral formula comes with a constant, \( C \), to account for any added constant that disappears upon differentiation.

Indefinite Integrals

An indefinite integral is the reverse operation of taking a derivative. Unlike definite integrals, which calculate the area under a curve over a specific interval, indefinite integrals represent a family of functions. When working with indefinite integrals like \( \int \sin x \, dx \), your goal is to find a function \( F(x) \) that, when differentiated, returns the integrand—in this case, \( \sin x \). The result of this process is the antiderivative, typically written as \( F(x) = -\cos x + C \).The "indefinite" part implies that there is no specific bounding value for \( x \); instead, the result is a general formula. This allows flexibility and includes a constant \( C \), representing an infinite number of possible antiderivatives. This is because any constant added to an indefinite integral's antiderivative will have its derivative equal to zero, hence disappearing upon taking the derivative.

Constant of Integration

The constant of integration is represented by \( C \) in the expression of an antiderivative or indefinite integral. This constant is crucial because it accounts for all possible constants that could have been added to the original function before differentiating it.Whenever you integrate a function, specifically an indefinite integral, you should include \( C \). This constant indicates the presence of an entire family of solutions, each differing by a constant. Why? Because the derivative of any constant is zero, and thus cannot be detected through differentiation. For example, in the problem involving \( \sin x \), its antiderivative is \( -\cos x + C \). The \( C \) here accounts for the fact that simply knowing the slope information (derivative) doesn't tell you where to start or stop along the vertical axis.It's a common mistake for beginners to forget the constant of integration, but understanding its role is fundamental to mastering calculus.