Problem 33 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the indefinite integrals. $$ \int \frac{(\ln x)^{2}}{x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{(\ln x)^{3}}{3} + C \).

Step 1: Identify the substitution

For the integral \( \int \frac{(\ln x)^{2}}{x} \ dx \), we notice the function \( (\ln x)^{2} \) and the other part \( \frac{1}{x} \). A good substitution can simplify the integral. Let \( u = \ln x \). This means \( du = \frac{1}{x} \, dx \), which matches the rest of the integrand.

Step 2: Substitute and transform the integral

Substituting \( u = \ln x \) into the integral, the expression becomes easier to solve. The integral \( \int \frac{(\ln x)^{2}}{x} \, dx \) transforms to \( \int u^{2} \, du \).

Step 3: Integrate with respect to the new variable

Now that we have \( \int u^{2} \, du \), apply the power rule for integration. The power rule states \( \int u^{n} \, du = \frac{u^{n+1}}{n+1} + C \) for \( n eq -1 \). Therefore, \( \int u^{2} \, du = \frac{u^{3}}{3} + C \).

Step 4: Substitute back the original variable

Convert back to the original variable using \( u = \ln x \). Thus, this integral transforms back to \( \frac{(\ln x)^{3}}{3} + C \).

Step 5: Write the final result

The evaluated indefinite integral is \( \int \frac{(\ln x)^{2}}{x} \ dx = \frac{(\ln x)^{3}}{3} + C \), where \( C \) represents the constant of integration.

Key Concepts

Indefinite Integrals

Indefinite integrals are all about finding the most general form of an antiderivative. When we perform indefinite integration, we are essentially working in the reverse direction of differentiation. The goal is to find a function whose derivative would give back the original function we started with.

One distinctive feature of indefinite integrals is the constant of integration, often represented as "C." This constant represents any number since the derivative of a constant is zero. For instance, if you integrate a function and get back an answer, adding any constant to it would still give the correct result.

In indefinite integration, knowing the basic derivatives and their antiderivatives is vital, since they guide us in the reverse process. It's helpful to remember these pairs to quickly and accurately evaluate integrals.

Integration Techniques

Integration techniques are the methods used to solve complex integrals, like substitution, integration by parts, and more. Each technique is particularly useful for different types of functions. For substitution, we look for a part of the integral that can be replaced with a single variable, simplifying the process.

Substitution Method:

• This method involves replacing a part of the integrand with a new variable.
• This replacement simplifies the integral into a standard form.
• After integration, substitute back to return to the original variable.

An essential step in substitution is identifying which part to choose for substitution. In the original exercise, the substitution was chosen as \( u = \ln x \). This choice made the integral much simpler, transforming it to \( \int u^{2} \, du \), which is straightforward to solve using the power rule.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. The natural logarithm has unique properties that make it very useful, especially in calculus. It appears frequently in integrals, derivatives, and limits.

Properties of Natural Logarithm:

• The derivative of \( \ln x \) is \( \frac{1}{x} \).
• The integral of \( \frac{1}{x} \) is \( \ln |x| \), assuming \( x eq 0 \).
• \( e^{\ln x} = x \).

In our original exercise, the natural logarithm is manipulated by substitution, demonstrating its flexibility in integration. With \( u = \ln x \), not only does the problem become easier to solve, but it also showcases the interplay between logarithmic and polynomial functions in integration processes.