Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Use the Log Rule to find the indefinite integral. $$ \frac{x}{x^{2}+4} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \(\frac{1}{2} ln |x^{2} + 4| + C\).

Step 1: Identify Suitable Substitution

Set \(u = x^{2} + 4\). It looks like a suitable substitution because its derivative \(2x\), up to a constant, is in the remaining part of the integrand.

Step 2: Find the Derivative of Substitution

Differentiate \(u\) w.r.t. \(x\) to get: \(du = 2x \, dx\). Divide both sides by \(2\) to find the value of \(dx\): \(dx = du/(2x)\).

Step 3: Substituting into the Integral

Substitute \(u\) and \(dx\) into the integral, and simplify the integral to obtain \(\frac{1}{2} \int \frac{1}{u} du\). This is a basic integral.

Step 4: Apply the Log Rule to Integrate

Now, integrate using the logarithmic rule \(\int \frac{1}{u} du = ln |u|\), to give \(\frac{1}{2} ln |u|\).

Step 5: Back-substitute the Original Variable

Substitute back \(u = x^{2} + 4\) to the answer to get \(\frac{1}{2} ln |x^{2} + 4|\). This is the indefinite integral.

Step 6: Add the Constant of Integration

Since the original problem is an indefinite integral, we have to add the constant of integration. The final answer becomes \(\frac{1}{2} ln |x^2 + 4| + C\), where \(C\) is the constant of integration.

Key Concepts

Integration by Substitution

In calculus, integration by substitution is a method often used to simplify a difficult integral into a more manageable form. The basic idea is to replace a part of the original integrand with a new variable, which simplifies the integral. One of the main challenges for students is identifying the right substitution.

In our exercise, we look for a substitution that will allow us to integrate more easily. We set \(u = x^2 + 4\), which is a suitable choice because its differential \(du = 2x \, dx\) is proportionate to the other factor in the integrand, \(x\). Dividing \(du\) by \(2\) gives us \(dx = \frac{du}{2x}\), which we can then substitute back into the integral. This step transforms the original integral into a simpler form where basic integration rules can be applied.

Logarithmic Integration

Logarithmic integration is a technique that deals specifically with integrals of the form \(\int \frac{1}{x} dx\), which is directly integrated to \(\ln|x| + C\), where \(C\) is the constant of integration.

The decision to use logarithmic integration in our exercise is made after substituting and simplifying the original integral to \(\frac{1}{2} \int \frac{1}{u} du\). The presence of \(\frac{1}{u}\) in the integral directly indicates the logarithmic rule applies. Through this method, we are able to integrate to find the natural logarithm of the absolute value of our substitution variable \(u\), which greatly simplifies the process of finding the indefinite integral.

Calculus

Calculus, the mathematical study of continuous change, plays a crucial role in understanding concepts such as derivatives and integrals. The area of calculus that deals with integrals is known as integral calculus.

In the context of the given exercise, integral calculus allows us to find the area under the curve of a function. By employing techniques such as integration by substitution, we can make complex problems simpler. Understanding the fundamentals of calculus is essential to grasp these techniques and to apply them effectively in various mathematical problems.

Constant of Integration

When dealing with indefinite integrals, the constant of integration is an essential aspect that represents the family of all antiderivatives. Whenever we integrate a function, we are finding all the possible functions that could differentiate to give us the original function. Because differentiation wipes out any constant, we must add a constant of integration, typically denoted as \(C\), to indicate those lost possibilities.

In our textbook problem, after applying logarithmic integration, we add the constant \(C\) to the final result to include all antiderivatives. It highlights a key point in calculus: an indefinite integral is not a single function, but a set of functions that differ by a constant.