91Ó°ÊÓ

In calculus, completing the square is a method used to simplify quadratic expressions. It helps to rewrite expressions so they appear as a perfect square, making them easier to integrate. Let's break this down step-by-step.
When given a quadratic expression like \(5 + 4x - x^2\), you want to rearrange it to make it a perfect square plus or minus a constant. First, rewrite the expression as \(-x^2 + 4x + 5\).
Factor out the negative sign from the quadratic part:\

• \(-1(x^2 - 4x) + 5\)

Next, within the parenthesis, apply the completing the square technique to the expression \(x^2 - 4x\).
Calculate \( \left(\frac{-4}{2}\right)^2 = 4 \) and adjust the expression to form a square:

• \( (x - 2)^2 - 4 \)

This lets us rewrite the entire expression as \( -((x - 2)^2) + 9 \), converting the original into a friendlier form, \( 9 - (x-2)^2 \).
Completing the square is instrumental in preparing the expression for further manipulation like substitution.

U-substitution is a technique used to simplify integrals, particularly when dealing with composite functions. It works by substituting a part of the integral with a new variable 'u'. This helps transform the original integral into a simpler one.
In the current scenario, after completing the square, set \( u = x - 2 \). This translates to \( x = u + 2 \). Additionally, the differential translates as \( dx = du \).
When substituting these into the integral \( \int \frac{x}{\sqrt{9 - (x-2)^2}} \, dx \), it becomes:

• \( \int \frac{u+2}{\sqrt{9 - u^2}} \, du \)

This splits into two separate integrals:

• \( \int \frac{u}{\sqrt{9-u^2}} \, du \)
• \( 2 \int \frac{1}{\sqrt{9-u^2}} \, du \)

U-substitution simplifies otherwise complex integrals, making them more manageable and easier to evaluate using further techniques.

Integral tables catalog solutions for common integrals, making it easier to solve integrals by providing formulas for specific integral types.In our problem, once we have split the main integral, each piece corresponds directly to integral forms found in the table:

• \( \int \frac{u}{\sqrt{a^2 - u^2}} \, du \) maps to \( -\sqrt{a^2 - u^2} + C \)
• \( \int \frac{1}{\sqrt{a^2 - u^2}} \, du \) maps to \( \arcsin\left(\frac{u}{a}\right) + C \)

The coefficients \(a=3\) are considered since the complete squared form was \((x-2)^2\).
So, the evaluation of our integral results in \(-\sqrt{9-u^2} + 2 \arcsin\left(\frac{u}{3}\right) + C\).
Integral tables are powerful tools in calculus that help quickly solve reoccurring integration problems, saving time and effort.

The arcsine function, represented as \( \arcsin(x) \), is the inverse of the sine function. It's used specifically when evaluating integrals that have a form like \( \int \frac{1}{\sqrt{a^2-u^2}} \, du \). In our problem, once the integrals have been separated and a suitable treatment is applied, the arcsine function appears as a natural solution.
In simplifying our expression, the part \(2 \int \frac{1}{\sqrt{9-u^2}} \, du\) results in:\

• \(2 \arcsin\left(\frac{u}{3}\right) + C \)

After all transformations and simplifications, replace back \(u = x - 2 \). Therefore,\

• \(2 \arcsin\left(\frac{x-2}{3}\right) + C \)

Arcsine helps in turning trigonometric functions back to a simpler form, making it essential in integration when handling specific square root terms.