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

User Jtm
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Answer:


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

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

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

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(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

Explanation:

Step 1: Define

Identify


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

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

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

Step 3: Integrate Pt. 2

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

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

Unit: Integration

User KorHosik
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