Mathematics, 15.07.2019 04:00 celestemaria0727

Evaluate the definite integral. use the integration capabilities of a graphing utility to verify your result. π/24 cos(12x) 0 dx

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Answer from: mayaa2351

$\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}$

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integrals

Integration Rule [Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx$

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

Set u: $\displaystyle u = 12x$ [u] Differentiate [Basic Power Rule, Derivative Properties]: $\displaystyle du = 12 \ dx$ [Bounds] Switch: $\displaystyle \left \{ {{x = \frac{\pi}{24} ,\ u = 12(\frac{\pi}{24}) = \frac{\pi}{2}} \atop {x = 0 ,\ u = 12(0) = 0}} \right$

Step 3: Integrate Pt. 2

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}\int\limits^{\frac{\pi}{24}}_0 {12\cos (12x)} \, dx$ [Integral] U-Substitution: $\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}\int\limits^{\frac{\pi}{2}}_0 {\cos (u)} \, du$ [Integral] Trigonometric Integration: $\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}[-sin(u)] \bigg| \limits^{\frac{\pi}{2}}_0$ Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}(1)$ Simplify: $\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}$

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

Unit: Integration

Answer from: Quest

honestly bro aint no one got time to answer these dumb question just do it p***y

step-by-step explanation:

Answer from: Quest

3-4

step-by-step explanation:

the answer is 3-4 because both 45 and 60 has a multiple of 15.

45/3=15

60/4=15

therefore the odds against selecting a red is 4-3

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