Problem 30 - Calculus Single Variable

Question

Integrate by parts successively to evaluate the given indefinite integral. $$ \int \ln ^{3}(x) d x $$

Step-by-Step Solution

Answer

\( \int \ln^3(x) \, dx = x\ln^3(x) - 3x\ln^2(x) + 6x\ln(x) - 6x + C \)

Step 1: Choose Components for Integration by Parts

Integration by parts is given by the formula: \( \int u \, dv = uv - \int v \, du \). Choose \( u = \ln^3(x) \) and \( dv = dx \) to start the integration process. This choice means that \( du = 3\ln^2(x) \cdot \frac{1}{x} dx \) and \( v = x \).

Step 2: Apply Integration by Parts

Now apply the integration by parts formula: \[ \int \ln^3(x) \, dx = x \ln^3(x) - \int x \cdot 3\ln^2(x) \cdot \frac{1}{x} \, dx = x\ln^3(x) - 3\int \ln^2(x) \, dx \]. This simplifies to \[ x\ln^3(x) - 3I_2 \], where \( I_2 = \int \ln^2(x) \, dx \).

Step 3: Integrate \( \int \ln^2(x) \, dx \) by Parts

Again, use integration by parts on \( I_2 \) with \( u = \ln^2(x) \) and \( dv = dx \). So \( du = 2\ln(x) \cdot \frac{1}{x} \, dx \) and \( v = x \). The formula becomes \( \int \ln^2(x) \, dx = x\ln^2(x) - \int x \cdot 2\ln(x) \cdot \frac{1}{x} \, dx = x\ln^2(x) - 2 \int \ln(x) \, dx \). This simplifies to \( x\ln^2(x) - 2I_1 \), where \( I_1 = \int \ln(x) \, dx \).

Step 4: Solve \( \int \ln(x) \, dx \)

For \( I_1 \), use integration by parts with \( u = \ln(x) \) and \( dv = dx \). Here, \( du = \frac{1}{x} dx \) and \( v = x \). The integration becomes \( \int \ln(x) \, dx = x\ln(x) - \int x \cdot \frac{1}{x} \, dx = x\ln(x) - x + C_1 \).

Step 5: Substitute Back the Intermediate Integrals

Substitute \( I_1 \) into the expression for \( I_2 \): \[ I_2 = x\ln^2(x) - 2(x\ln(x) - x + C_1) = x\ln^2(x) - 2x\ln(x) + 2x - 2C_1 \].

Step 6: Final Integration by Parts Solution

Substitute \( I_2 \) back into the original integration expression: \[ \int \ln^3(x) \, dx = x\ln^3(x) - 3(x\ln^2(x) - 2x\ln(x) + 2x - 2C_1) = x\ln^3(x) - 3x\ln^2(x) + 6x\ln(x) - 6x + C_2 \].

Step 7: Add Constants of Integration

Combine constants \( C \) by letting \( C = C_2 + 6C_1 \). The final expression, including the constant of integration, is: \[ \int \ln^3(x) \, dx = x\ln^3(x) - 3x\ln^2(x) + 6x\ln(x) - 6x + C \].

Key Concepts

Understanding Indefinite Integrals

An indefinite integral represents the collection of all antiderivatives of a function, essentially reversing the process of differentiation. When integrating a function, no limits of integration are specified, which implies that constants of integration are added.
To express the result of an indefinite integral, we often add a constant `C`, which accounts for any vertical shift possible in the graph of an antiderivative. In the original scenario, where we dealt with \( \int \ln^3(x) \, dx \), the process involves several steps of integration by parts.

• The integration approach was initiated without specified limits which is characteristic of indefinite integrals.
• Finding the antiderivative of a composite function such as \( \ln^3(x) \) requires decomposition into simpler parts, which can be successively integrated by parts.
• The result is not a single function, but instead all functions, differing by a constant, derived from this integration.

Understanding the indefinite integral facilitates navigating through complex integrals like the one presented, and involves comprehending the interaction of functions being integrated.

Natural Logarithm Integration in Depth

Integrating natural logarithm functions such as \( \ln(x) \) is a frequent scenario in calculus, commonly requiring integration by parts. Because logarithmic functions grow slower than polynomial functions, integrating them presents unique challenges.
The natural logarithm function \( \ln(x) \) often appears in composite forms, like \( \ln^3(x) \), needing careful handling during integration.
The integration by parts technique is particularly useful here. We use the formula:
\[ \int u \, dv = uv - \int v \, du \]
Here, by choosing \( u \) as a natural log term like \( \ln^3(x) \), we focus on gradually reducing its power through repetitive application of the method. This results in simpler sub-integrals, each one involving powers of the natural logarithm descending with each step.
Being familiar with this logarithmic characteristic simplifies seemingly intricate integrals by splitting them into manageable pieces through technique and practice.

Exploring the Repeated Integration Process

The repeated integration process involves tackling complex integrals by breaking them down into simpler, more manageable integrals. Through successive applications of integration by parts, each step simplifies part of the integral until a straightforward result is achievable.
The original task shows a need for repeated integration because \( \ln^3(x) \) is a high power of a logarithmic function. Handling it requires multiple rounds of integration by parts, each one reducing the power of \( \ln(x) \) until the integral itself becomes manageable.
In each iteration:

• The choice of \( u \) is vital, typically taken as the portion of the integrand that simplifies upon differentiation, in this case, \( \ln^3(x) \), \( \ln^2(x) \), and so forth.
• As each integration cycle reduces the complexity, it turns the integral into a simpler one, allowing us to eventually obtain a complete antiderivative.
• The recursive breakdown ensures all terms are accounted for, always finished by adding a constant of integration to finalize the indefinite integral.

Through this method, we've effectively managed the integral of \( \ln^3(x) \) in smaller steps, clearly demonstrating the utility and necessity of repeated integration by parts in calculus problem-solving.