\int\limits^0_∞ cos{x} \, dx

General Formulas and Concepts:

Pre-Calculus

Unit CircleTrig Graphs

Calculus

LimitsLimit Rule [Variable Direct Substitution]:                                                                 IntegralsIntegration Rule [Fundamental Theorem of Calculus 1]:                                            Trig IntegrationImproper Integrals

Step-by-step explanation:

Step 1: Define

Identify

Step 2: Integrate

[Improper Integral] Rewrite:                                                                             [Integral] Trig Integration:                                                                                [Integral] Evaluate [Integration Rule - FTC 1]:                                                  Evaluate trig:                                                                                                     Evaluate limit [Limit Rule - Variable Direct Substitution]:                              

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.

∴

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

Unit: Improper Integrals

Book: College Calculus 10e