Problem 14 - Calculus for Biology and Medicine

Question

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\frac{x}{1+x} $$

Step-by-Step Solution

Answer

The general antiderivative is \( x - \ln|1+x| + C \).

Step 1: Identify the Function

The function we are given to find the general antiderivative for is \( f(x) = \frac{x}{1+x} \). To proceed, we notice that it is a rational function which might be suitable for algebraic manipulation before integration.

Step 2: Simplify the Function

Rewriting the function \( \frac{x}{1+x} \) by splitting the numerator can make integration easier. We recognize that this can be rewritten as \( 1 - \frac{1}{1+x} \) because \( \frac{x}{1+x} = 1 - \frac{1}{1+x} \).

Step 3: Integrate the Simplified Function

Integrate the function \( 1 - \frac{1}{1+x} \). The integral of \( 1 \) is \( x \), and the integral of \( \frac{1}{1+x} \) is \( \ln|1+x| \) due to the natural logarithm rule for integration.

Step 4: Write the General Antiderivative

Combine the results from the integrations: \[\int \left(1 - \frac{1}{1+x} \right)\, dx = \int 1\, dx - \int \frac{1}{1+x} \, dx = x - \ln|1+x| + C\]Here, \( C \) is the constant of integration representing the family of all antiderivatives.

Key Concepts

Understanding Rational Functions

A rational function is a type of function represented by the ratio of two polynomials. Specifically, a rational function can be expressed in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). These functions play a crucial role in calculus as they often require specific techniques for integration due to their fraction form.Understanding how to handle rational functions involves knowing how to simplify them. Simplification can include polynomial long division, factoring, or even splitting the function as seen in the given problem:

• Identifying common factors in the numerator and the denominator can help reduce the complexity of the function.
• Algebraic manipulation, like rewriting \( \frac{x}{1+x} \) as \( 1 - \frac{1}{1+x} \), allows for easier integration.

By breaking down complex expressions into simpler parts, integration becomes more manageable. This insight is vital for successfully finding antiderivatives of rational functions.

Basics of Integration

Integration is a fundamental concept in calculus, often seen as the reverse of differentiation. It involves finding the antiderivative of a function, which means determining the original function given its derivative. Integration can be definite, which finds the area under a curve, or indefinite, which focuses on antiderivatives.To integrate a function, we aim to find a function whose derivative gives us the original function. The general process involves:

• Recognizing patterns or using known integral rules, like \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for different powers of \( x \).
• Breaking complex expressions into simpler parts that are easier to integrate, using algebraic manipulation or substitution.
• Applying foundational rules, such as the power rule, exponential rule, or trigonometric integrals accordingly.

Overall, mastering the integration process requires practice and familiarity with various integral techniques. This process enables students to solve a wide range of mathematical problems by finding general antiderivatives.

Natural Logarithm in Integration

The natural logarithm, denoted as \( \ln(x) \), is an essential concept in calculus related to integration. Specifically, when integrating functions of the form \( \frac{1}{x} \), the antiderivative is \( \ln|x| + C \). This rule derives from the fundamental property that the derivative of \( \ln(x) \) is \( \frac{1}{x} \).Utilizing this rule becomes particularly useful when dealing with rational functions, where the integrand can often be rearranged to fit this form. For instance, in the given problem, the function part \( \frac{1}{1+x} \) integrates to \( \ln|1+x| \). This gives:

• Simplicity in handling logarithmic integrals, which simplifies calculations involving logs.
• Understanding that \( |x| \) is used because the domain of the natural logarithm is limited to positive values, ensuring solutions have broad applicability.

Comprehending how the natural logarithm interacts with integration not only facilitates solving complex problems but also deepens comprehension of the integral's relationship to logarithmic functions.