Problem 13 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=\frac{1}{1+x} $$

Step-by-Step Solution

Answer

The general antiderivative of \( f(x) = \frac{1}{1+x} \) is \( \ln|1+x| + C \).

Step 1: Identify the Integral to Solve

The function given is \( f(x) = \frac{1}{1+x} \). We need to find the antiderivative of this function, which is essentially finding the indefinite integral \( \int \frac{1}{1+x} \, dx \).

Step 2: Recognize the Form of the Function

The expression \( \frac{1}{1+x} \) is in the form \( \frac{1}{1+u} \) where \( u = x \). The antiderivative of \( \frac{1}{1+u} \) with respect to \( u \) is known to be the natural logarithm function \( \ln|1+u| + C \), where \( C \) is the constant of integration.

Step 3: Apply the Antiderivative Formula

Using the formula from the previous step, since \( u=x \), the antiderivative of \( f(x) = \frac{1}{1+x} \) is \( \ln|1+x| + C \).

Step 4: State the Final General Antiderivative

Thus, the general antiderivative of the given function is \( F(x) = \ln|1+x| + C \), where \( C \) is the constant of integration.

Key Concepts

Antiderivative

The antiderivative is a fundamental concept in integral calculus, serving as the reverse process of differentiation. When we speak of finding an antiderivative for a function, what we are really doing is determining a function whose derivative will produce the original function we started with. Let’s say you have a function \( f(x) \), the antiderivative of this function is another function, often denoted as \( F(x) \), such that \( F'(x) = f(x) \).

• The function \( F(x) \) is called the antiderivative or the primitive of \( f(x) \).
• If \( F(x) \) is an antiderivative of \( f(x) \), then any function of the form \( F(x) + C \) is also an antiderivative, where \( C \) is an arbitrary constant.

Understanding antiderivatives is critical because they allow us to reverse the process of differentiation, thus finding the original function before it was derived.

Indefinite Integral

The indefinite integral is closely tied to the concept of the antiderivative. In fact, looking for an indefinite integral is essentially finding antiderivatives. When we calculate the indefinite integral of a function \( f(x) \), we usually represent it as:
\[ \int f(x) \, dx = F(x) + C \]
This notation signifies the collection of all possible antiderivatives of \( f(x) \). The integral sign \( \int \) is an invitation to find all possible functions \( F(x) \) whose derivative will give you back the initial function \( f(x) \).

• The process of finding the indefinite integral is also called integration, and it involves using rules and formulas to work backwards from the derivative back to the original function.
• The "+ C" part of the expression is crucial because it represents the constant of integration, acknowledging that there are infinite such antiderivatives for any function.

Whenever you find an indefinite integral, remember it signifies finding a general form of antiderivatives.

Natural Logarithm

The natural logarithm is denoted by \( \ln \) and is a specific logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. In calculus, the natural logarithm is significant because it often appears as the result of integrating functions in a particular form.

• A common integration formula is: \( \int \frac{1}{1 + u} \, du = \ln|1 + u| + C \). This reveals how integrals involving expressions like \( \frac{1}{1 + u} \) are naturally linked to the function \( \ln|1 + u| \).
• In the context of integration, the absolute value around \( 1 + u \) in \( \ln|1 + u| \) ensures we account for the domain where the logarithm is defined.

The natural logarithm has powerful applications in solving integrals because it directly relates to the fundamental antiderivatives we often deal with in integral calculus.