Problem 77 - University Calculus: Early Transcendentals

Question

Verify the formulas in Exercises by differentiation. $$\int \frac{1}{x+1} d x=\ln |x+1|+C, \quad x \neq-1$$

Step-by-Step Solution

Answer

The formula is correct as the derivative of \( \ln|x+1| + C \) equals \( \frac{1}{x+1} \).

Step 1: Understand the Task

We need to verify that the given integral formula is correct by differentiating the right-hand side expression and ensuring that it equals the original integrand.

Step 2: Write Down the Formula

The formula provided is \( \int \frac{1}{x+1} dx = \ln |x+1| + C \). We aim to differentiate the expression \( \ln |x+1| + C \) with respect to \( x \).

Step 3: Differentiate the Right-Hand Side

We differentiate \( \ln |x+1| + C \). The derivative of \( \ln |x+1| \) for \( x eq -1 \) is \( \frac{1}{x+1} \) because the derivative of \( \ln u \) is \( \frac{1}{u} \times \frac{du}{dx} \) and for \( u = x+1 \), \( \frac{d(x+1)}{dx} = 1 \). The derivative of a constant \( C \) is 0.

Step 4: Compare with Original Integrand

The derivative \( \frac{1}{x+1} \) matches the original integrand \( \frac{1}{x+1} \). Since the differentiated result is equal to the original integrand, the formula is verified as correct.

Key Concepts

Integral Verification

Integral verification is an essential step in calculus to confirm that an integral expression is correctly formulated. It primarily involves differentiating the right-hand side of the integral and comparing it to the original integrand. This ensures that applying the fundamental theorem of calculus gives the proper integral.

In our example, we have the integral \( \int \frac{1}{x+1} \, dx = \ln |x+1| + C \). To verify, we differentiate \( \ln |x+1| + C \) and check if it equals the original integrand, \( \frac{1}{x+1} \):

• Differentiate the function \( \ln |x+1| \).
• Result in \( \frac{1}{x+1} \), which is identical to our integrand.

This match confirms that our integral formula is correct. Such verifications are crucial to confirm our calculus results are accurate.

Differentiation Steps

Understanding the process of differentiation is key to solving calculus problems. The differentiation of functions helps us find slopes and rate of changes, which are vital concepts in calculus.

When differentiating, follow these steps:

• Identify the function you need to differentiate. Here, \( \ln |x+1| + C \).
• Recognize that \( \ln |x+1| \) involves a composite function where \( u = x+1 \).
• Apply the chain rule: the derivative of \( \ln u \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).

Since \( \frac{d(x+1)}{dx} = 1 \), the derivative of \( \ln |x+1| \) becomes \( \frac{1}{x+1} \). Remember, the derivative of constant \( C \) is zero, so it plays no role in your result.

Logarithmic Derivatives

Logarithmic derivatives come into play when you're working with functions involving logarithms. They simplify the differentiation process by transforming multiplication into addition, making it easier to work with complex products or quotients.

For the natural logarithm function \( \ln u \):

• The derivative is \( \frac{1}{u} \cdot \frac{du}{dx} \).
• Useful for functions such as \( \ln |x+1| \), where \( u = x+1 \).

Using this principle, when we differentiate \( \ln |x+1| \), or generally any log function, we first find the derivative of the inside function, then multiply it by the reciprocal of the inside function. This not only confirms the derivation process but also helps in integral verification, as we observe in the exercise. By understanding logarithmic derivatives, students can more easily tackle a variety of calculus problems, ensuring their grasp on this fundamental topic is robust.