Problem 17 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=\frac{1}{1+2 x} $$

Step-by-Step Solution

Answer

The antiderivative is \( F(x) = \frac{1}{2} \ln |1+2x| + C \).

Step 1: Recognize the Integral

The given function is \( f(x) = \frac{1}{1+2x} \), which is a rational function. We can solve it using a basic formula for integrals of the form \( \frac{1}{ax+b} \).

Step 2: Use the Basic Integral Formula

The integral of \( \frac{1}{ax+b} \) is \( \frac{1}{a} \ln|ax+b| + C \), where \( C \) is the constant of integration. This is derived from the rule of integration for logarithmic functions.

Step 3: Identify Constants

In the function \( \frac{1}{1+2x} \), \( a = 2 \) and \( b = 1 \). These constants will be used in our formula to find the antiderivative.

Step 4: Apply the Formula

Substitute \( a = 2 \) and \( b = 1 \) into the formula: \( \int \frac{1}{1+2x} \, dx = \frac{1}{2} \ln |1+2x| + C \).

Step 5: Write the General Antiderivative

The general antiderivative of \( f(x) = \frac{1}{1+2x} \) is \( F(x) = \frac{1}{2} \ln |1+2x| + C \), where \( C \) is an arbitrary constant.

Key Concepts

Rational Function

A rational function is any function that can be written as the quotient of two polynomials. In simpler terms, it's a fraction where the numerator and the denominator are both polynomial expressions. The function from the exercise, \( f(x) = \frac{1}{1+2x} \), is a classic example. Rational functions often include expressions like \( \frac{x^2 + 3}{x - 2} \), or even simpler forms like our exercise.

When integrating rational functions, particularly when the degree of the polynomial in the numerator is less than that in the denominator, certain methods and formulas are useful. Recognizing the structure of the rational function helps in choosing the right integration formula or technique to find the antiderivative. In our case, the rational function has a constant numerator of "1", which simplifies the integration process using the integral formula for logarithms.

Integral Formula

The integral formula used to find the antiderivative of \( \frac{1}{ax+b} \) is:

• \( \int \frac{1}{ax+b} \ dx = \frac{1}{a} \ln |ax+b| + C \)

This formula is derived from the fundamental rule of integrating logarithmic functions and is particularly handy for functions with a linear denominator. The presence of \( ax + b \) in the denominator suggests a natural logarithmic relationship in its antiderivative.

In the exercise, \( a \) and \( b \) are constants extracted from \( ax+b \). By identifying \( a = 2 \) and \( b = 1 \) in our function, we plug these into the integral formula, leading to our solution. The term \( \frac{1}{a} \) scales the natural logarithm expression, ensuring the correct size of the antiderivative. This scaling is necessary because of the product rule in differentiation — accounting for the multiplication of \( x \) by \( a \).

Constant of Integration

The constant of integration, represented as \( C \), is a crucial concept when finding a general antiderivative. An antiderivative is not unique; it represents a family of functions that differ by a constant. This is because when you differentiate any constant, the derivative is zero. Thus, the "+ C" accounts for all these constants.

In our solution, \( F(x) = \frac{1}{2} \ln |1+2x| + C \), the \( C \) ensures that any vertical shift in the function is considered. In other words, it's the most general way to represent the indefinite integral, allowing for any possible values that result from an initial condition when given.

Understanding the constant of integration helps in solving real-world problems involving integration, as it allows you to ascertain the correct particular solution when additional information, like a point on the graph, is available.