Question

For the following exercises, find the definite or indefinite integral. $$ \int \frac{(\ln x)^{2} d x}{x} $$

Short answer

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

Step-by-step solution

Identify the Type of Integral and Use Substitution

We are given the integral \( \int \frac{(\ln x)^2}{x} \, dx \). This is an indefinite integral involving a natural logarithm. To solve it, we can use the substitution method. Let \( u = \ln x \). Then, the differential \( du = \frac{1}{x} dx \), or equivalently, \( dx = x \, du \).

Transform the Integral with Substitution

Substituting \( u = \ln x \) into the integral, we get \( \int u^2 \, du \), because \( \frac{(\ln x)^2}{x} dx = u^2 du \). The integral is now much simpler to solve.

Integrate the Transformed Integral

Integrate \( \int u^2 \, du \) using the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Here, \( n = 2 \), so \( \int u^2 \, du = \frac{u^3}{3} + C \).

Substitute Back to the Original Variable

Replace \( u \) with \( \ln x \) to express the integral in terms of \( x \). Therefore, \( \frac{u^3}{3} + C \) becomes \( \frac{(\ln x)^3}{3} + C \).

Write the Final Answer

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

Key concepts

Indefinite Integral

When dealing with calculus problems, you'll often come across the term "indefinite integral." An indefinite integral is a form of integration without specified limits, which essentially means that it is the opposite operation of differentiation. Instead of finding a precise numerical value, indefinite integrals provide a family of functions that represent the accumulated area under a curve. The beauty of indefinite integrals is that they always include a constant of integration, denoted as "+ C." This constant represents all possible vertical shifts of the anti-derivative function along the y-axis, due to the absence of specific limits.

• Symbol: The indefinite integral is represented by the integral sign \( \int \), followed by the function to integrate and the differential "dx" or "du".
• Result: The result includes the original function's antiderivative plus the constant of integration \( C \).

Understanding the purpose of indefinite integrals helps us make sense of the fundamental principles underlying calculus, linking rates of change back to accumulated quantities.

Substitution Method

The substitution method is a helpful technique for solving integrals, especially when the integral is complex or involves functions that a direct integration isn't easy for. The core idea behind this method is to simplify the integrand (the function being integrated) by introducing a new variable and transforming the integral into a simpler form. This often involves substituting a part of the function with a single variable, chosen to simplify the expression.
In the exercise, you were given \( \int \frac{(\ln x)^2}{x} \, dx \). Here, you let \( u = \ln x \), making the differential \( du = \frac{1}{x} \, dx \). This simplifies the original integral so it becomes \( \int u^2 \, du \) after substitution.

• Step 1: Choose the substitution \( u \), making the integral easier to handle.
• Step 2: Compute the differential \( du \), using the relationship between \( u \) and \( dx \).
• Step 3: Replace the original variable with \( u \) in the integral to simplify and solve.

Substituting back into the original variable at the end is crucial to express the solution in the context of the original problem. The substitution method is a powerful tool in integration and crucial for tackling complex integrals in calculus.

Integration of Logarithmic Functions

Integrating logarithmic functions can initially seem challenging, but with the right techniques, it becomes an approachable task. Logarithmic functions are prevalent in calculus, representing slow growth and decay across numerous applications. When encountering integrals involving logarithms, substitution is often the go-to method due to the logarithm's derivative properties.
For the integral \[ \int \frac{(\ln x)^2}{x} \, dx \], integrating directly isn't straightforward due to the composition of logarithms and exponentiation. Substitution was used to simplify it, letting \( u = \ln x \) convert the integral into a polynomial form \( \int u^2 \, du \). Here, the power rule can be easily applied, providing a clearer path to the solution.

• Logarithmic Integration Techniques: Focus on transforming logarithmic functions into manageable algebraic expressions.
• Power Rule Application: After substitution, often the simplified integral can be solved using the power rule \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \).
• Substitute Back: It's vital to reintroduce the original variables to express the solution in the correct context.

Mastering integration of logarithmic functions involves recognizing when and how to apply substitution to reduce complexity, leading to an effective solution strategy.