Question

What is the largest possible integral value in the domain of the real-valued function

Answer 1

Answer:

Max Value: x = 400

General Formulas and Concepts:

Algebra I

• Domain is the set of x-values that can be inputted into function f(x)

Calculus

• Antiderivatives
• Integral Property: $\int {cf(x)} \, dx = c\int {f(x)} \, dx$
• Integration Method: U-Substitution
• [Integration] Reverse Power Rule: $\int {x^n} \, dx = \frac{x^{n+1}}{n+1} + C$

Step-by-step explanation:

Step 1: Define

$f(x) = \frac{1}{\sqrt{800-2x} }$

Step 2: Identify Variables

Using U-Substitution, we set variables in order to integrate.

$u = 800-2x\\du = -2dx$

Step 3: Integrate

1. Define:                                                                                                            $\int {f(x)} \, dx$
2. Substitute:                                                                                         $\int {\frac{1}{\sqrt{800-2x} } } \, dx$
3. [Integral] Int Property:                                                                                     $-\frac{1}{2} \int {\frac{-2}{\sqrt{800-2x} } } \, dx$
4. [Integral] U-Sub:                                                                                           $-\frac{1}{2} \int {\frac{1}{\sqrt{u} } } \, du$
5. [Integral] Rewrite:                                                                                          $-\frac{1}{2} \int {u^{-\frac{1}{2} }} \, du$
6. [Integral - Evaluate] Reverse Power Rule:                                                 $-\frac{1}{2}(2\sqrt{u}) + C$
7. Simplify:                                                                                                         $-\sqrt{u} + C$
8. Back-Substitute:                                                                                            $-\sqrt{800-2x} + C$
9. Factor:                                                                                                           $-\sqrt{-2(x - 400)} + C$

Step 4: Identify Domain

We know from a real number line that we cannot have imaginary numbers. Therefore, we cannot have any negatives under the square root.

Our domain for our integrated function would then have to be (-∞, 400]. Anything past 400 would give us an imaginary number.