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sweet [91]
3 years ago
6

\int\limits^0_∞ cos{x} \, dx

Mathematics
1 answer:
JulijaS [17]3 years ago
7 0

Answer:

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

General Formulas and Concepts:

<u>Pre-Calculus</u>

  • Unit Circle
  • Trig Graphs

<u>Calculus</u>

  • 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:

<u>Step 1: Define</u>

<em>Identify</em>

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

<u>Step 2: Integrate</u>

  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

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