Problem 41 - Calculus for Biology and Medicine

Question

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} d x $$

Step-by-Step Solution

Answer

\( \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} \, dx = \tan^{-1}(g(x)) + C \).

Step 1: Identify the Substitution

Given the integral \( \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} \, dx \), notice that the term \( [g(x)]^2 + 1 \) resembles a derivative that would arise from the arc tangent function. Thus, set \( u = g(x) \), which implies \( du = g^{\prime}(x) \, dx \).

Step 2: Rewrite the Integral in Terms of \( u \)

Substitute \( u = g(x) \) and \( du = g^{\prime}(x) \, dx \) into the integral, so that it becomes \( \int \frac{1}{u^2 + 1} \, du \).

Step 3: Identify the Integral Form

Recognize that \( \int \frac{1}{u^2 + 1} \, du \) is a standard integral form, which is equal to \( \tan^{-1}(u) + C \), where \( C \) is the constant of integration.

Step 4: Substitute Back to \( x \)

Replace \( u \) with \( g(x) \) to get the final result: \( \tan^{-1}(g(x)) + C \).

Key Concepts

Substitution Method

The substitution method is a powerful technique in integral calculus designed to simplify complex integrals. It works by changing variables to make integration more straightforward. In this context, you replace the original variable with another one that reduces the complexity of the integrand.

Here's how it works in simple terms:

• Identify a part of the integrand (the function you want to integrate) that can be substituted with a single variable, often labeled as "\( u \)".
• This substitution is usually chosen to simplify the integral's algebraic expression, making it look more like a standard integral form.
• The derivative of the part you choose is used to replace the differential expression (e.g., \( dx \)), effectively changing the variable of integration.

By transforming one variable into another, you're shifting the problem into a more familiar form. This allows you to use standard integral results, easing the calculation process. This strategy was employed in the exercise by setting \( u = g(x) \) allowing the integral to simplify, making it easier to identify a known integral form.

Arc Tangent Function

The arc tangent function, often written as \( \tan^{-1}(x) \) or \( \text{arctan}(x) \), is the inverse of the tangent function. In the context of integration, it's recognized by certain standard integral forms.

One particular standard form is:

• \( \int \frac{1}{x^2 + 1} \, dx = \tan^{-1}(x) + C \)

This relationship arises because the derivative of \( \tan^{-1}(x) \) is \( \frac{1}{x^2 + 1} \). When you encounter an integral with this form, it directly leads to the arc tangent function as a solution.

In the exercise, once the substitution was made, the integral transformed into \( \int \frac{1}{u^2 + 1} \, du \). This aligns perfectly with the arc tangent function, hence the additional simplification to \( \tan^{-1}(u) + C \), and then replacing \( u \) back with \( g(x) \). This highlights the importance of recognizing these integral forms for efficient problem-solving.

Indefinite Integral

An indefinite integral in calculus represents a family of functions whose derivative is the given function. It is expressed without upper and lower limits and includes an arbitrary constant \( C \) because any constant added to a function disappears upon differentiation.

Here's what you need to know:

• The process of finding an indefinite integral is known as anti-differentiation or integration.
• The result of an indefinite integral is a general solution, containing an arbitrary constant \( C \), representing an infinite set of functions.
• The goal of integration is to "reverse" differentiation, hence why it's called the anti-derivative.

In the exercise, the aim was to find the indefinite integral of a complex function. Using substitution, it simplified to a form where known integral results, such as those involving the arc tangent function, could be applied. The final solution \( \tan^{-1}(g(x)) + C \) represents all possible functions that have the given function as their derivative. This encapsulates the essence of an indefinite integral.