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Integration by parts is a handy method often used when dealing with integrals involving a product of two functions. It is derived from the product rule for differentiation and provides a way to transform a complex integral into simpler parts. The formula is:

• \( \int u \, dv = uv - \int v \, du \)

The idea is to break down the integration of a product into two simpler integrals, one of which can be directly evaluated.
When using integration by parts, you first need to identify parts of the function in the form of \( u \) and \( dv \). You then find \( du \) by differentiating \( u \) and integrate \( dv \) to find \( v \). This allows you to replace a difficult integral with expressions that are easier to manage.
In our specific example, our choice was \( u = \ln x \) and \( dv = x^2 \, dx \), leading to relatively straightforward derivative and integral results.

A definite integral calculates the accumulation of certain quantities over a specific interval on a function. It is denoted by \( \int_{a}^{b} f(x) \, dx \), with \( a \) and \( b \) being the lower and upper bounds of integration, respectively.
The definite integral results in a number, which represents the net "area" between the function and the x-axis within the specified limits.

• The process typically involves first finding the indefinite integral, often termed as the antiderivative.
• Once the antiderivative is found, the values of the upper and lower limits are then substituted into this antiderivative.
• Finally, subtraction of these values provides the result of the definite integral.

In this exercise, we need to evaluate the definite integral of \( x^{2} \ln x \) over the range from \( e \) to \( e^2 \), with proper evaluations at these boundaries.

Logarithmic functions, exemplified by \( \ln x \), are one of the fundamental types of functions in calculus. Logarithms have a base, and in this case, the natural logarithm \( \ln x \) uses base \( e \).

• The derivative of the natural logarithm is simple: \( \frac{d}{dx} \ln x = \frac{1}{x} \).
• This property makes logarithmic functions very convenient to work with when doing calculus.

Logarithms often appear in integration scenarios where they can drastically simplify the process.
In integration by parts, the natural logarithm's straightforward derivative facilitates calculating \( du \). Thus, choosing \( u = \ln x \) makes the integration process more efficient, turning complex integrals into solvable expressions.

Selecting the right function for \( u \) and \( dv \) in integration by parts can be crucial. A common strategy is using the ILATE rule, which prioritizes choosing:

• I: Inverse functions (like \( \text{arctan} \; x \))
• L: Logarithmic functions (like \( \ln x \))
• A: Algebraic functions (like polynomials \( x^2 \))
• T: Trigonometric functions (like \( \sin x, \cos x \))
• E: Exponential functions (like \( e^x \))

This mnemonic suggests choosing \( u \) based on which function appears first on this list, prioritizing logarithmic or algebraic expressions when possible.
In our problem \( \int x^2 \ln x \, dx \), choosing \( \ln x \) as \( u \) lends well to differentiation because its derivative \( \frac{1}{x} \) simplifies the integration process. Meanwhile, \( x^2 \) becomes \( dv \) due to its straightforward integration to \( \frac{x^3}{3} \). Employing this approach makes the integration by parts formula effective and manageable.