Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int \frac{x}{3 x^{2}-1} d x$$

Step-by-Step Solution

Answer

The short answer is: \(\int \frac{x}{3 x^{2}-1} d x = \frac{1}{6} \ln|3x^2-1| + C\)

Step 1: Identify the proper substitution

A good substitution candidate is the function \(u = 3x^2 - 1\). This choice will simplify the integrand.

Step 2: Find the differential of the substitution function

Now we differentiate the substitution function with respect to \(x\). Using the power rule of differentiation, we have: \[du = \frac{d}{dx}(3x^2 - 1) = 6x\,dx\] Notice that the result contains the term \(x\,dx\) which appears in the original integrand.

Step 3: Make the substitution and solve for \(x\,dx\)

Now we rewrite the original integral in terms of \(u\). First, we solve for \(x\,dx\) in the expression for the differential \(du\): \[x\,dx = \frac{1}{6}\,du\]

Step 4: Substitute and integrate

Substitute \(u\) and \(x\,dx\) into the original integral and integrate with respect to \(u\): \[\int \frac{x}{3x^2 - 1} \,dx = \int \frac{1}{u} \cdot \frac{1}{6}\,du = \frac{1}{6}\int \frac{1}{u}\,du\] Now, we integrate the function \(\frac{1}{u}\) with respect to \(u\). The integral of \(\frac{1}{u}\) is simply the natural logarithm: \[\frac{1}{6}\int \frac{1}{u}\,du = \frac{1}{6}\ln|u| + C\]

Step 5: Substitute back the original variable

Finally, replace \(u\) with the original expression in terms of \(x\): \[\frac{1}{6}\ln|u| + C = \frac{1}{6}\ln|3x^2 - 1| + C\] Thus, the indefinite integral of the given function is: \[\int \frac{x}{3 x^{2}-1} d x = \frac{1}{6} \ln|3x^2-1| + C\]

Key Concepts

Substitution Method

The substitution method is a powerful tool for evaluating integrals, making complex expressions more manageable. Think of it as changing variables to simplify the problem at hand.
In this method, you select a part of the integral to replace with a new variable, commonly called "u." Once this substitution is chosen, you differentiate it, creating a new differential term that corresponds to the original integral.

• Choose the substitution: A good substitution candidate helps simplify the integral. In our case, we chose \(u = 3x^2 - 1\) because it appears in the denominator, simplifying the integration process significantly.
• Differentiate to find \(du\): Here, take the derivative of \(u = 3x^2 - 1\). This gives \(du = 6x \, dx\). Notice the presence of \(x \, dx\), which is part of the initial integral.
• Solve for \(x \, dx\): Rearrange the equation \(du = 6x \, dx\) to find \(x \, dx = \frac{1}{6} \, du\). This allows you to directly replace terms in the original integral.

By applying the substitution method, you turn a complicated integral into a simpler one, paving the way for easier integration.

Natural Logarithm

The natural logarithm, often denoted as \(\ln\), is a fundamental concept in calculus, commonly appearing in integral solutions. It is the inverse function of the exponential function \(e^x\), which means that it "undoes" the effect of the exponential function.
In integration, the integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln|u| + C\), where \(C\) is the constant of integration. This form is especially prevalent in integration problems involving rational functions.

• In our problem, after substitution, the integral simplifies to \(\int \frac{1}{u} \, du\), representing a classic scenario of integrating to the natural logarithm.
• Once integrated, we have \(\ln|u|\), indicative of the presence of the natural logarithm in our result.
• Lastly, remember that \(|u|\) denotes the absolute value, ensuring the argument of the \(\ln\) function remains positive, which is crucial in real-number mathematics.

Understanding the role of natural logarithms in integration helps demystify how solutions evolve and provides deeper insight into the behaviors of functions as they relate to their integrals.

Differentiation

Differentiation involves finding the derivative of a function. It measures how a function's output changes concerning changes in its input. It is the foundational tool for understanding rates of change and is the reverse of integration.
In our exercise, differentiation helps determine the necessary substitution terms in the integration process. For the substitution method, differentiating the chosen \(u\) function is essential.

• Begin with \(u = 3x^2 - 1\): Differentiation results in \(du = 6x \, dx\). This differentiation gives us the needed term that helps replace \(x \, dx\) in the integral.
• Apply the power rule: When differentiating expressions like \(3x^2\), use the power rule, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\). Here, it transforms \(3x^2\) into \(6x\).
• Relating back to integration: Differentiation and integration are inverse processes. By differentiating the chosen substitution, we set the stage for the integration process to simplify and solve the integral.

Grasping differentiation is crucial as it empowers students to handle various calculus challenges, especially when paired with integration techniques like the substitution method.