Question

\int\limits^0_∞ cos{x} \, dx

Answer 1

Answer:

$\displaystyle \int\limits^0_\infty {cos(x)} \, dx = sin(\infty)$

General Formulas and Concepts:

Pre-Calculus

• Unit Circle
• Trig Graphs

Calculus

• Limits
• Limit Rule [Variable Direct Substitution]:                                                     $\displaystyle \lim_{x \to c} x = c$
• Integrals
• Integration Rule [Fundamental Theorem of Calculus 1]:                             $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$
• Trig Integration
• Improper Integrals

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^0_\infty {cos(x)} \, dx$

Step 2: Integrate

1. [Improper Integral] Rewrite:                                                                         $\displaystyle \lim_{a \to \infty} \int\limits^0_a {cos(x)} \, dx$
2. [Integral] Trig Integration:                                                                             $\displaystyle \lim_{a \to \infty} sin(x) \bigg| \limits^0_a$
3. [Integral] Evaluate [Integration Rule - FTC 1]:                                               $\displaystyle \lim_{a \to \infty} sin(0) - sin(a)$
4. Evaluate trig:                                                                                                 $\displaystyle \lim_{a \to \infty} -sin(a)$
5. Evaluate limit [Limit Rule - Variable Direct Substitution]:                           $\displaystyle -sin(\infty)$

Since we are dealing with infinity of functions, we can do a numerous amount of things:

• Since -sin(x) is a shift from the parent graph sin(x), we can say that -sin(∞) = sin(∞) since sin(x) is an oscillating graph. The values of -sin(x) already have values in sin(x).
• Since sin(x) is an oscillating graph, we can also say that the integral actually equates to undefined, since it will never reach 1 certain value.

∴  $\displaystyle \int\limits^0_\infty {cos(x)} \, dx = sin(\infty) \ or \ \text{unde}\text{fined}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Improper Integrals

Book: College Calculus 10e