Problem 21 - Calculus Early Transcendentals

Question

Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$G(s)=\frac{1}{s^{2}+1}$$

Step-by-Step Solution

Answer

Question: Find all the antiderivatives of the function G(s) = (1)/(s^2 + 1) and verify the result by taking the derivatives. Answer: The antiderivatives of the function G(s) = (1)/(s^2 + 1) are given by F(s) = tan^(-1)(s) + C, where C is a constant.

Step 1: Identify the function whose derivative matches the given function

We can notice that the given function is related to the derivative of tan\(^{-1}\)(s) or arctan(s), as the derivative of arctan(s) is \(\frac{1}{s^2+1}\).

Step 2: Write down the antiderivative

As G(s) represents the derivative of arctan(s), we can write the antiderivative F(s) as: $$F(s) =\int G(s)\, ds = \int \frac{1}{s^2+1}\,ds = \tan^{-1}(s) + C$$ Here, C is the integration constant, which is added because the antiderivative is not unique.

Step 3: Verify the result by taking the derivative

To check if our solution is correct, we will find the derivative of F(s). $$\frac{dF}{ds} = \frac{d}{ds}\left(\tan^{-1}(s) + C\right)$$ Using the chain rule and remembering that \(\frac{d}{ds}\tan^{-1}(s)=\frac{1}{s^2+1}\), we get: $$\frac{dF}{ds}=\frac{1}{s^2+1}$$ We observe that the derivative of F(s) matches with the given function G(s), so our solution for the antiderivative is correct. The antiderivatives of G(s) are given by: $$F(s) = \tan^{-1}(s) + C$$

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change. It's subdivided into two areas: differential calculus, which concerns itself with rates of change and slopes of curves; and integral calculus, which focuses on the accumulation of quantities and the area under and between curves.

Understanding calculus is fundamental to many scientific disciplines, including physics, engineering, economics, statistics, and beyond. The principals of calculus are applied to model real-world systems, optimize performance, and solve complex problems that involve variables changing with respect to one another.

Integral Calculus

Integral calculus is concerned with finding the function when its derivative is known, this process is known as anti-differentiation or integration. The two main types of integrals are indefinite and definite integrals. An indefinite integral represents a family of functions and includes an integration constant to account for all possible antiderivatives. On the other hand, a definite integral computes the net area under a curve over a certain interval, providing a specific value.

Integral calculus is imperative in calculating volumes, areas, displacement, and other concepts that arise by summing infinitesimal data points.

Indefinite Integral

An indefinite integral represents all the antiderivatives of a given function. It's written in the form of \(\int f(x)\,dx\), which is interpreted as the general antiderivative of f(x) plus a constant C, known as the integration constant. This constant is crucial because there's an infinite number of functions that can have the same derivative, hence the 'indefinite' quality of this integral, symbolizing that it's not limited to just one solution.

The process of finding an indefinite integral is often called integration or anti-differentiation and requires familiarity with a variety of integration techniques and rules to simplify and solve integrals effectively.

Arctan Integration

Integration involving the arctangent function, or \(\text{arctan} x\), is a common task in calculus. Since the derivative of \(\text{arctan} x\) is \(\frac{1}{x^2+1}\), it plays a role in the integration of functions that involve \(\frac{1}{x^2+1}\) in their denominator.

To integrate a function whose antiderivative involves arctan, like \(\int \frac{1}{s^2+1}\,ds\), one must recognize that this is the derivative of \(\text{arctan} s\) and the indefinite integral \(\int \frac{1}{s^2+1}\,ds = \text{arctan} s + C\), where C is the integration constant. Effectively, recognizing when to apply arctan integration is a key skill in solving such integral problems.

Integration Constant

The integration constant, typically denoted as C, is a crucial aspect of indefinite integrals. Since the act of differentiation of a function cancels any added constant, when we find the antiderivative, we must include this constant to account for all possible original functions. For example, if a function F(x) is differentiated to f(x), integrating f(x) back to its antiderivative should yield F(x) plus any real number constant C, as all would have the same derivative f(x).

In practical terms, whenever you solve an indefinite integral, always append the '+ C' to your solution, acknowledging that there's an entire family of functions that could potentially satisfy the given integral. It is the difference between finding 'a' solution and 'the general' solution to a given problem.