Problem 8 - Calculus Single Variable

Question

Integrate by parts to evaluate the given indefinite integral. $$ \int \ln (x) x^{2} d x $$

Step-by-Step Solution

Answer

\( \int \ln(x) x^{2} \, dx = \frac{x^3}{3} \ln(x) - \frac{x^3}{9} + C \).

Step 1: Identify components for integration by parts

The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \). We should choose \( u \) and \( dv \) from the integrand. In this case, let \( u = \ln(x) \) and \( dv = x^2 \, dx \). We chose \( u \) as \( \ln(x) \) to make differentiation simpler.

Step 2: Differentiate and Integrate

Differentiate \( u \) to find \( du \), and integrate \( dv \) to find \( v \):- \( du = \frac{1}{x} \, dx \) because the derivative of \( \ln(x) \) is \( \frac{1}{x} \).- \( v = \frac{x^3}{3} \) because the integral of \( x^2 \) is \( \frac{x^3}{3} \).

Step 3: Apply the integration by parts formula

Plug \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula:\[ \int \ln(x) x^{2} \, dx = \left( \ln(x) \cdot \frac{x^3}{3} \right) - \int \frac{x^3}{3} \cdot \frac{1}{x} \, dx \]Simplify the expression inside the integral:\[ \int \frac{x^3}{3} \cdot \frac{1}{x} \, dx = \frac{1}{3} \int x^2 \, dx \]

Step 4: Evaluate the remaining integral

Evaluate \( \frac{1}{3} \int x^2 \, dx \):\[ \frac{1}{3} \int x^2 \, dx = \frac{1}{3} \cdot \frac{x^3}{3} = \frac{x^3}{9} \]Now substitute back into the equation from Step 3:\[ \int \ln(x) x^{2} \, dx = \frac{x^3}{3} \ln(x) - \frac{x^3}{9} \]

Step 5: Simplify the expression

Combine the terms to simplify:\[ \int \ln(x) x^{2} \, dx = \frac{x^3}{3} \ln(x) - \frac{x^3}{9} + C \] Where \( C \) is the constant of integration. This is the final result.

Key Concepts

Indefinite Integral

In calculus, an indefinite integral is essentially the reverse process of differentiation. It is a function that describes the accumulation of quantities, like areas under curves, without defined limits. When you solve an indefinite integral, you're seeking a function whose derivative is the integrand, the function inside the integral sign.
The result of an indefinite integral is a general solution that includes a constant of integration, typically denoted by "C." This constant represents all the possible vertical shifts of the antiderivative since differentiation eliminates any constant.

• The example problem involves the indefinite integral of a function combining a logarithmic and a polynomial expression: \[ \int \ln(x) x^{2} \, dx \]
• With no specific interval given, the task is to find the function whose derivative corresponds to this integrand.
  

The integration by parts technique, derived from the product rule of differentiation, can be used here to assist in solving these more complex integrals.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change or the slope of the function at any given point. In the context of integration by parts, differentiation is used alongside integration to solve integrals that involve products of functions. Here's how it fits into our exercise.
To apply integration by parts, you first identify the parts of the integrand that will be differentiated and integrated. In our example, choosing:

• \( u = \ln(x) \) because its differentiation is straightforward. The derivative is: \[ du = \frac{1}{x} \, dx \]
• \( dv = x^2 \, dx \) making it simpler to integrate.

These choices simplify the problem because differentiating \( \ln(x) \) reduces the complexity, while the polynomial \( x^2 \) integrates easily.
Remember that differentiation reduces the power of a polynomial or simplifies a complicated function, enabling easier integration by parts.

Logarithmic Function

Logarithmic functions, typically represented as \( \ln(x) \), have unique properties that make them essential in solving certain integrals. In this exercise, \( \ln(x) \) is strategically chosen as part of the integration by parts method due to these properties.
Some important features of logarithmic functions include:

• They are defined for positive values of \( x \) and approach infinity as \( x \) increases.
• The derivative of \( \ln(x) \) simplifies significantly to \( \frac{1}{x} \).

In the example, selecting \( u = \ln(x) \) leverages the effective differentiation property, converting a potentially complicated integral into a solvable problem. When faced with products involving logs and polynomials, it is advantageous to apply these unique logarithmic functions' characteristics within integration by parts.
This choice aids in isolating complex components and streamlining integration challenges.

Polynomial Integration

Polynomial integration is one of the more straightforward techniques in calculus, often forming the other integral component within integration by parts problems. The fundamental process involves applying the power rule for integration: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] This principle is crucial when integrating the polynomial part of our exercise.
In the given problem:

• We chose \( dv = x^2 \, dx \). Integrating this gives:\[ v = \frac{x^3}{3} \]
• The simpler structure is key to effectively applying integration by parts.

Here, the polynomial \( x^2 \) lends itself well to integration because, when paired with differentiation of \( \ln(x) \), it combines to simplify the integral significantly.
Understanding polynomial integration enables handling various terms in an expression, allowing them to reduce into manageable components, a fundamental skill when tackling more advanced integration techniques.