Problem 44 - Calculus for Biology and Medicine

Question

Assume that a is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{a}{a+x} $$

Step-by-Step Solution

Answer

The general antiderivative is \( a \ln |a + x| + C \).

Step 1: Identify the type of function

The function given is \( f(x) = \frac{a}{a+x} \). This is a rational function and can be approached using logarithmic integration, as the derivative of the denominator is a constant multiple of the numerator.

Step 2: Substitute variables for integration

Let \( u = a + x \). Consequently, the derivative is \( du = dx \). This substitution allows us to rewrite the integrand as \( \frac{a}{u} \).

Step 3: Setup the integration

With the substitution \( u = a + x \) and \( du = dx \), the integral becomes \( \int \frac{a}{u} \, du \) which can be expressed as \( a \int \frac{1}{u} \, du \).

Step 4: Integrate using logarithmic rule

The integral of \( \int \frac{1}{u} \, du \) is \( \ln |u| + C \), where \( C \) is the constant of integration. Thus, the integral becomes \( a \ln |u| + C \).

Step 5: Substitute back to original variables

Replace \( u \) back with \( a + x \). This gives your antiderivative as \( a \ln |a + x| + C \).

Key Concepts

Understanding Antiderivative

An antiderivative, often called the indefinite integral, is a function that reverses the process of differentiation. In simple terms, if you have a function and you differentiate it, finding the antiderivative will bring you back to the original function. For a given function, there can be many antiderivatives, differing by a constant, typically denoted as "C." This constant arises because the derivative of a constant is zero, and thus does not affect the function's derivative.

To find an antiderivative, you need to determine a function whose derivative is equal to the function you started with. In calculus, finding antiderivatives is central to solving many real-world problems involving areas and accumulations.

• The integral sign \( \int \) denotes finding an antiderivative.
• The function being integrated is known as the integrand.
• Antiderivatives are related to definite integrals by the Fundamental Theorem of Calculus.

Recognizing when and how to find antiderivatives is crucial, especially to resolve problems involving physics and engineering scenarios.

Rational Functions Demystified

Rational functions are fractions where the numerator and the denominator are polynomials. These functions often appear in calculus problems because they model various real-life systems effectively. In our given function, \( f(x) = \frac{a}{a+x} \), it is a rational function because both the top (numerator) and bottom (denominator) can be considered polynomial expressions.

Here's what you need to know:

• Rational functions can have asymptotes, which are lines that the graph approaches but never touches. These can be vertical, horizontal, or slant.
• Simplifying rational functions can reveal more about the function's behavior and make integration easier.
• In integration, recognizing a rational function can simplify your approach, especially when the derivative of the denominator is proportional to the numerator.

For our exercise, the presence of a rational function hints at potential methods for integration, guiding us towards using substitution and logarithmic rules.

Logarithmic Integration Explored

Logarithmic integration is a technique used in calculus when dealing with integrals of the form \( \int \frac{1}{x} \, dx \), which integrates to \( \ln |x| + C \). This method is especially useful when you're handling functions where the derivative of one part (like the denominator) matches the form of the numerator.

In the exercise, we used logarithmic integration after substituting \( u = a + x \), transforming our integral into a simpler form. Here's a brief step-by-step:

• Identify if the integrand aligns with \( \int \frac{1}{x} \, dx \).
• Use substitution where necessary to simplify the integrand.
• Apply the logarithmic integration rule to solve the integral.
• Don't forget to add the constant of integration, \( C \).
• Once integrated, revert back to the original variables to express your solution fully.

This technique is a staple in calculus, especially when facing functions similar to \( \frac{a}{a+x} \), and understanding it broadens your toolkit for handling integrals efficiently.