Question

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

Answer 1

`\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