Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Use the Log Rule to find the indefinite integral. $$ \int \frac{1}{x+1} d x $$

Step-by-Step Solution

Answer

So the integral \(\int \frac{1}{x+1} dx\) equals \(ln|x+1| + C\).

Step 1: Identify the integral

Firstly, the integral given in the exercise is \(\int \frac{1}{x+1} dx\). This function is in the form of a simple logarithmic integral rule, where \(\int \frac{f'(x)}{f(x)} dx = log|f(x)|\). Here the function \(f(x)\) is \(x+1\).

Step 2: Apply the Log Rule

The next step is to apply the Log Rule over the given integral. The Log Rule says that the integral of \(dx/(x+1)\) is \(ln|x+1|\). After applying the Log Rule, the integral becomes the logarithm of the absolute value of \(x+1\).

Step 3: Add the constant of integration

Because we are finding the indefinite integral, a constant of integration will be added. The constant of integration is usually represented by \(C\). So the integral \(\int \frac{1}{x+1} dx\), becomes \(ln|x+1| + C\).

Key Concepts

Logarithmic Integration

Logarithmic integration is a method used in calculus for integrating rational functions. Here, the variable appears in the denominator in such a way that it fits into a natural logarithm function. This technique primarily revolves around the Log Rule, which is incredibly handy when we encounter integrals of the form \( \int \frac{f'(x)}{f(x)} dx \).
The Log Rule states that this integral results in \( \ln |f(x)| + C \), where \( C \) is the constant of integration.
Let's break it down:

• Identify the function \( f(x) \) whose derivative \( f'(x) \) appears in the numerator.
• Ensure that \( \frac{1}{f(x)} \) is in the integrand, forming a clear natural logarithm pattern.
• Apply the Log Rule directly, integrating surprisingly faster once you spot this form.
• Finally, don't forget to add the constant of integration, \( C \), to complete the integral.

Recognizing this pattern allows you to use logarithmic integration with ease and precision.

Constant of Integration

When calculating indefinite integrals, we always add a constant of integration, denoted as \( C \). This arises because integration is basically the reverse process of differentiation. While derivatives eliminate constant terms, integrals do not have a mechanism to acknowledge constants not directly derivable back.
Therefore, when reversing derivatives in form of integration, multiple functions might share the same derivative but vary by a constant. For example, \( \ln |x+1| \), \( \ln |x+1| + 2 \), or even \( \ln |x+1| - 7 \) all differentiate to the same value. Therefore, when dealing with indefinite integrals:

• The resultant function must contain an arbitrary constant \( C \) to cover all potential original functions.
• Failing to include \( C \) would result in incomplete solutions.

Always remember: adding \( C \) completes your solution.

Calculus Integration Rules

Calculus integration rules form the backbone of solving integrals efficiently. These rules help us transform complex integrals into more manageable forms, paving the way for quick solutions. Some key rules to remember involve:

• **Power Rule**:\( \int x^n dx = \frac{x^{n+1}}{n+1} + C; n eq -1 \)
• **Exp Rule**: \( \int e^x dx = e^x + C \)
• **Constant Rule**: \( \int a \, dx = ax + C \)
• **Additive Rule**: \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \)
• **Scalar Multiplication**: \( \int a \cdot f(x) \, dx = a \cdot \int f(x) \, dx \)

These rules are essential tools in solving various integration problems, including logarithmic forms. With practice, identifying which rule to apply becomes intuitive, leading to streamlined problem-solving in calculus.