Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral. $$ \int x^{2} e^{3 x} d x $$

Step-by-Step Solution

Answer

\[u = x^{2}\] and \(d v = e^{3 x} dx\).

Step 1: Identify the function \(u\)

According to the rule of LIATE, an Algebraic function \(u\) is chosen before an Exponential function. Here, the Algebraic function is \(x^{2}\), so set \(u = x^{2}\).

Step 2: Identify the function \(d v\)

The remaining function in the integrand will be \(d v\). In this case, it is \(e^{3 x} dx\), so set \(d v = e^{3 x} dx\).

Key Concepts

Integral Calculus

Integral calculus is one of the two main branches of calculus, the other being differential calculus. While differential calculus focuses on the rate of change of functions, integral calculus deals with the accumulation of quantities and the areas under and between curves. The primary tool used in integral calculus is the integral, which can be thought of as the opposite of a derivative.
The goal in integral calculus is often to find the antiderivative of a function. This involves determining a function which derivative is equal to the original function we started with. There are various methods to solve integrals, with integration by parts being one useful technique.
Integration by parts is particularly handy for integrating products of functions. It is based on the formula:\[\int u \, dv = uv - \int v \, du\]where we choose the parts of the integrand to assign to \( u \) and \( dv \). This method is a powerful tool for tackling complex integrals that cannot be solved easily using simple rules of integration, pivoting on algebraic manipulation and substitution.

Algebraic Functions

Algebraic functions are mathematical expressions constructed using variables and constants, combined with operations such as addition, subtraction, multiplication, division, and taking roots. The expression \(x^2\) found in our original integral is an algebraic function.
The choice of our algebraic function \(u\) in the integration by parts method follows the LIATE rule, which is a mnemonic device representing the order of preference for choosing \(u\) based on the type of function:

• Logarithmic functions
• Inverse trigonometric functions
• Algebraic functions
• Trigonometric functions
• Exponential functions

In our integral, \(u = x^2\) is selected first because it is an algebraic function, making subsequent derivatives easier to handle. Always opting for the function that becomes simpler upon differentiation makes the integration process smoother.

Exponential Functions

Exponential functions involve expressions where a constant base is raised to a variable exponent. In our task, the part \(e^{3x}\) of the integral represents an exponential function.
Exponential functions grow or decay at rates proportional to their current value, making them crucial in modeling growth processes in sciences and economics. In integration by parts, when combined with algebraic functions, exponential functions often make for straightforward differentiation and are chosen as \(dv\) after following the LIATE rule.
For our integral \( \int x^2 e^{3x} \, dx\), \(dv = e^{3x} \, dx\) because the derivative and antiderivative of an exponential function are relatively easier to calculate. The natural exponential function's consistent growth pattern simplifies the differentiation process, ensuring we can handle more complex expressions in the integral smoothly.