Problem 299 - Calculus Volume 1

Question

For the following exercises, find the indefinite integral. $$\int \frac{d x}{1+x}$$

Step-by-Step Solution

Answer

\( \ln|1+x| + C \), where \( C \) is the constant of integration.

Step 1: Identify the Integral Form

The given integral is \( \int \frac{d x}{1+x} \). This can be recognized as a standard form for which the antiderivative is known. This form is typically encountered when integrating functions that produce a natural logarithm.

Step 2: Recall the Antiderivative Formula

The integral of \( \frac{1}{x + a} \) is \( \ln|x+a| + C \), where \( C \) is the constant of integration. In our case, \( a = 1 \), which aligns perfectly with the integral \( \int \frac{1}{1+x} \, dx \).

Step 3: Apply the Antiderivative Formula

Applying the formula, the indefinite integral \( \int \frac{1}{1+x} \, dx \) becomes \( \ln|1+x| + C \).

Step 4: Simplify and State the Final Answer

Express the integration result clearly. Thus, the indefinite integral of \( \frac{1}{1+x} \) is \( \ln|1+x| + C \), where \( C \) remains an arbitrary constant.

Key Concepts

Antiderivative

An antiderivative is essentially the reverse process of differentiation. Imagine you have a function, say \( f(x) \), and you are given its derivative. The antiderivative of this derivative would be the original function that, when differentiated, results in \( f(x) \). In a way, it's like retracing steps in a mathematical dance to find out where you started. In our exercise, the function given for integration is \( \frac{1}{1+x} \). Using knowledge of standard forms, it's understood that the antiderivative of this function involves natural logarithms because it matches the pattern of functions that differentiate into \( \frac{1}{x} \) or \( \frac{1}{1+x} \). This makes recognizing forms very useful.

Knowing how to spot these forms and their antiderivatives can speed up integration, saving you from more complex arithmetic or calculus nitty-gritty. It's kind of like having shortcuts in math that allow you to think ahead and execute with ease.

Natural Logarithm

The natural logarithm, denoted \( \ln \), is a fundamental concept in calculus often popping up in integration problems, particularly with forms like \( \frac{1}{x} \). It's based on the constant \( e \), approximately equal to 2.718, which is important in many growth models like populations or investments and decay models, which include radioactive decay.

In the given exercise, \( \ln|1+x| \) appears as the result of the indefinite integral of \( \frac{1}{1+x} \). This is because the integral\( \int \frac{1}{1+x} \, dx = \ln|1+x| + C \) follows directly from the antiderivative formula. The natural logarithm here provides a solution that makes sense when considering how areas beneath the curve of \( \frac{1}{1+x} \) grow. Understanding this relationship helps students recognize the significance of natural logarithms beyond simple arithmetic and connects it to the real-world representations and continuous exponential growth.

Constant of Integration

When dealing with indefinite integrals, you're always adding something called the constant of integration, expressed as \( C \). Why do we add \( C \)? Simply put, the process of differentiation loses information about constants. Practically, this means any constant added to a function disappears when you take its derivative. When finding an antiderivative, this information needs to be accounted for, thus the introduction of \( C \).

In our example, the indefinite integral of \( \frac{1}{1+x} \) is \( \ln|1+x| + C \). Here, \( C \) represents an infinite collection of potential vertical shifts of the curve \( \ln|1+x| \), since it doesn't alter the shape, only the position on a graph. This concept reemphasizes that indefinite integrals reveal a family of functions, not just a single solution, offering a fuller picture of possibilities in mathematical solutions.