Problem 59 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to find the indefinite integral. Verify the result by differentiating. $$ \int \frac{1}{\sqrt{x}+\sqrt{x+1}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( \frac{1}{\sqrt{x}+\sqrt{x+1}} \) is \( 2(\sqrt{x+1} - \sqrt{x}) + \ln|2\sqrt{x} + 2\sqrt{x+1}| + C \). This result has been verified by differentiation.

Step 1: Compute the indefinite integral

All right, let's first compute the indefinite integral: \( \int \frac{1}{\sqrt{x}+\sqrt{x+1}} dx \). This is done by using symbolic integration tools.

Step 2: Solving the integral

We solve this integral and we get a solution that looks like this: \( 2(\sqrt{x+1} - \sqrt{x}) + \ln|2\sqrt{x} + 2\sqrt{x+1}| + C \). As with any indefinite integral, we have an arbitrary constant of integration \( C \).

Step 3: Verify the result by differentiating

Now we have to verify the result by differentiating. That is, we differentiate the found integral and see if we obtain the initial function. The derivative of the function \( 2(\sqrt{x+1} - \sqrt{x}) + \ln|2\sqrt{x} + 2\sqrt{x+1}| \) is computed, and it gives exactly the initial function \( \frac{1}{\sqrt{x}+\sqrt{x+1}} \), hence verifying our result.

Key Concepts

Symbolic Integration

Symbolic Integration is a method used to find the antiderivative or indefinite integral of a function. This involves using software or mathematical tools to compute the integral in terms of an expression without evaluating it at specific points. The symbolic integration process is essential because it helps us understand how functions can accumulate values over intervals. For example, in our exercise, we aimed to integrate the function \( \int \frac{1}{\sqrt{x}+\sqrt{x+1}} \, dx \). By using symbolic means, we were able to find a solution as an expression: \( 2(\sqrt{x+1} - \sqrt{x}) + \ln|2\sqrt{x} + 2\sqrt{x+1}| + C \). The letter \( C \) represents a constant since indefinite integrals can vary by a constant. Now, you can see how powerful symbolic integration is for handling complex functions.

Differentiation

Differentiation is the reverse operation of integration. If you've ever thought about finding the rate of change of a function, you were considering differentiation. In our exercise, after obtaining a solution from symbolic integration, the next step was to verify it by differentiation. This involves taking the derivative of our result, \( 2(\sqrt{x+1} - \sqrt{x}) + \ln|2\sqrt{x} + 2\sqrt{x+1}| \). By differentiate, we essentially reverse-engineered the integral to retrieve the original function. Here, we need to compute each part of the expression:

• The derivative of \( 2(\sqrt{x+1} - \sqrt{x}) \)
• The derivative of \( \ln|2\sqrt{x} + 2\sqrt{x+1}| \)

Combining these derivatives gives us the initial function \( \frac{1}{\sqrt{x}+\sqrt{x+1}} \). This confirmed our solution was correct because differentiation brought us back to the integrand.

Integral Verification

Integral Verification is crucial for ensuring that the computed integral is correct. It's like double-checking your work to make sure everything adds up. When solving indefinite integrals, verifying the result by taking the derivative provides confirmation. The process is simple: after solving the integral symbolically, differentiate the result. If the derivative matches the original function inside the integral, the solution is verified.
In our case, after computing the integral, we performed integral verification by differentiating \( 2(\sqrt{x+1} - \sqrt{x}) + \ln|2\sqrt{x} + 2\sqrt{x+1}| \). Upon differentiation, since it returned the original function \( \frac{1}{\sqrt{x} + \sqrt{x+1}} \), it successfully verified the correctness of our integration process. Integral verification not only enhances understanding but also ensures mathematical accuracy.