Problem 21 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{(\ln x)^{2}}{x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the function \(\frac{(\ln x)^{2}}{x}\) with respect to \(x\) is \(\frac{1}{3}(\ln x)^{3} + C\).

Step 1: Making a Substitution

Let's make the substitution \(u = \ln x\). The fact that \(x = e^u\) derives from this. As a result, the derivative of \(x\) with respect to \(u\), or \(dx\), is \(e^u du\).

Step 2: Substituting \(\ln x\) and \(dx\) in the integral

We replace \(\ln x\) with \(u\) and \(dx\) with \(e^u du\) in the integral. That gives us: \(\int \frac{u^{2}}{e^u} e^u du\) or \(\int u^2 du\).

Step 3: Solving the Integral

The integral \(\int u^{2} du\) is a standard one. Applying power rule of integration, it becomes \(\frac{1}{3}u^{3} + C\), where \(C\) is the constant of integration.

Step 4: Returning to the Original Variable

The last step is substituting \(u\) back for \(\ln x\) which gives the final solution as \(\frac{1}{3}(\ln x)^{3} + C\).

Key Concepts

Substitution Method

The substitution method is a powerful tool in calculus, especially for solving indefinite integrals. It involves substituting a part of the integral with a new variable, which simplifies the integral into a form that's easier to solve.

Consider our exercise, where the integral \( \int \frac{(\ln x)^{2}}{x} dx \) seems complex at first. By making the substitution \( u = \ln x \) and recognizing that \( x = e^u \) as a consequence, the derivative of \( x \) with respect to \( u \) becomes \( dx = e^u du \). This clever change of variables simplifies the integral significantly.

After the substitution, the integral becomes \( \int u^2 du \) which is much easier to solve using the power rule. The key is to always remember to back-substitute the original variable to complete the solution, as demonstrated in the step-by-step solution provided.

Power Rule of Integration

The power rule of integration is essential for solving integrals of polynomial functions. The rule states that to integrate a power of \( u \), say \( u^n \), where \( n \) is a real number and not equal to -1, you add 1 to the exponent and divide by the new exponent, adding the constant of integration at the end.

In the context of our example, after the substitution, we apply the power rule to \( u^2 \), and integrate to get \( \frac{1}{3}u^3 \). It's important to apply this rule correctly to arrive at the correct antiderivative. For students aiming to deepen their understanding, practicing various power integrals and remembering that the rule does not apply when \( n = -1 \) (as this would lead to a logarithmic integral instead), is highly beneficial.

Constant of Integration

When determining an indefinite integral, a critical concept to include is the constant of integration, denoted as \( C \). This embodies the idea that the antiderivative of a function is not unique; there is actually an infinite family of functions that differ only by a constant.

In our exercise, after integrating \( \frac{1}{3}u^3 \) using the power rule, adding \( C \) is necessary to account for all possible antiderivatives. The existence of \( C \) respects the fact that the derivative of a constant is zero, and thus when we reverse the process of differentiation through integration, we must accommodate all potential constants that were originally present. Remembering to add \( C \) is vital for the correctness and completeness of the solution.