Problem 52 - Calculus and its applications

Question

Evaluate. $$ \int x^{2}(\ln x)^{2} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{x^{3}}{3} (\ln x)^{2} - \frac{2x^{3}}{9} \ln x + \frac{2x^{3}}{27} + C \).

Step 1: Identify the Integration Method

The integral \( \int x^{2} (\ln x)^{2} \ dx \) involves a polynomial and a logarithmic function. This is a good candidate for integration by parts, a technique used for integrating products of functions.

Step 2: Choose Parts for Integration by Parts

Integration by parts follows the formula: \( \int u\ dv = uv - \int v\ du \). We choose \( u = (\ln x)^{2} \) and \( dv = x^{2} \ dx \).

Step 3: Compute du and v

Find the differential \( du \) by differentiating \( u \):\[ du = 2(\ln x) \cdot \frac{1}{x} \ dx = \frac{2(\ln x)}{x} \ dx \]Find \( v \) by integrating \( dv \):\[ v = \int x^{2} \ dx = \frac{x^{3}}{3} \]

Step 4: Apply Integration by Parts Formula

Substitute \( u, du, v, dv \) into the integration by parts formula:\[ \int x^{2} (\ln x)^{2} \ dx = \frac{x^{3}}{3} (\ln x)^{2} - \int \left( \frac{x^{3}}{3} \right) \frac{2(\ln x)}{x} \ dx \] This simplifies to:\[ \frac{x^{3}}{3} (\ln x)^{2} - \frac{2}{3} \int x^{2} \ln x \ dx \]

Step 5: Apply Integration by Parts Again

Now, evaluate \( \int x^{2} \ln x \ dx \) using integration by parts by choosing \( u = \ln x \) and \( dv = x^{2} \ dx \).Find \( du \):\[ du = \frac{1}{x} \ dx \]Find \( v \):\[ v = \frac{x^{3}}{3} \]

Step 6: Compute Second Part

Using these new components in the formula:\[ \int x^{2} \ln x \ dx = \frac{x^{3}}{3} \ln x - \int \frac{x^{3}}{3} \cdot \frac{1}{x} \ dx \]This simplifies to:\[ \frac{x^{3}}{3} \ln x - \frac{1}{3} \int x^{2} \ dx \] Which evaluates as:\[ \frac{x^{3}}{3} \ln x - \frac{1}{9} x^{3} + C \]

Step 7: Combine Results

Substitute this back into the expression from Step 4:\[ \int x^{2} (\ln x)^{2} \ dx = \frac{x^{3}}{3} (\ln x)^{2} - \frac{2}{3} \left( \frac{x^{3}}{3} \ln x - \frac{1}{9} x^{3} \right) \]Distribute the \( -\frac{2}{3} \):\[ \int x^{2} (\ln x)^{2} \ dx = \frac{x^{3}}{3} (\ln x)^{2} - \frac{2x^{3}}{9} \ln x + \frac{2x^{3}}{27} + C \]

Step 8: Finalize the Solution

The evaluated expression for the integral is:\[ \int x^{2} (\ln x)^{2} \ dx = \frac{x^{3}}{3} (\ln x)^{2} - \frac{2x^{3}}{9} \ln x + \frac{2x^{3}}{27} + C \]Here, \( C \) is the constant of integration.

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of calculus that deals with the concept of integration. Integration is essentially the reverse process of differentiation. It helps us find areas under curves, among many other applications. In the context of our exercise, we're looking at the integral of a function that combines a polynomial and a logarithmic component.

Integrals can be definite or indefinite:

• **Definite integrals** are calculated over a specific interval and give a numeric result representing the area under a curve within that interval.
• **Indefinite integrals**, like the one in our exercise, include a constant of integration and represent a family of functions.

Choosing the right method for integration is crucial. Integration by parts, as used in our exercise, is an essential technique for dealing with products of functions that don't easily lend themselves to basic integration. By strategically choosing parts of the function, this technique derives a complicated integral from simpler ones.

Polynomial Integration

Polynomial integration involves integrating expressions that include polynomial terms, which are powers of a variable, typically denoted by 'x'. These types of functions are generally straightforward to integrate. For instance, the power rule for integration states that:

• To integrate a polynomial expression like \( x^n \), where \( n eq -1 \), you can apply: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

In our exercise, we encounter\( x^2 \) as part of the integral. Integrating \( x^2 \) gives us \( \frac{x^3}{3} + C \). Yet, due to the presence of a logarithmic term, we cannot directly apply this and must use integration by parts.

Remember that polynomial functions can sometimes be mixed with other functions, causing complexity, but the power rule remains a cornerstone of polynomial integration.

Logarithmic Functions

Logarithmic functions have forms like \( \ln(x) \), and they often appear in more complex integration problems due to their non-polynomial nature. In our integration by parts problem, one of the parts we used was \( (\ln x)^2 \).

While differentiating logarithmic functions, recall that:

• The derivative of \( \ln(x) \) is \( \frac{1}{x} \).
• For more complex forms such as \( (\ln x)^2 \), use the chain rule to differentiate effectively. This requires extra steps but is manageable.

During integration by parts, the presence of a logarithm means carefully selecting terms for differentiation. Choosing correctly can simplify the integral significantly. In our exercise, identifying \( u = (\ln x)^2 \) was key to breaking down the integral effectively. This way, the problem becomes more manageable when applying the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \].

Logarithmic functions might seem intimidating but, with these strategies, they become part of a more systematic approach to solving integrals.