Answer:
[tex]\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
Integration Rule [Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
U-Substitution
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx[/tex]
Step 2: Integrate Pt. 1
Identify variables for u-substitution.
- Set u: [tex]\displaystyle u = 12x[/tex]
- [u] Differentiate [Basic Power Rule, Derivative Properties]: [tex]\displaystyle du = 12 \ dx[/tex]
- [Bounds] Switch: [tex]\displaystyle \left \{ {{x = \frac{\pi}{24} ,\ u = 12(\frac{\pi}{24}) = \frac{\pi}{2}} \atop {x = 0 ,\ u = 12(0) = 0}} \right[/tex]
Step 3: Integrate Pt. 2
- [Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}\int\limits^{\frac{\pi}{24}}_0 {12\cos (12x)} \, dx[/tex]
- [Integral] U-Substitution: [tex]\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}\int\limits^{\frac{\pi}{2}}_0 {\cos (u)} \, du[/tex]
- [Integral] Trigonometric Integration: [tex]\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}[-sin(u)] \bigg| \limits^{\frac{\pi}{2}}_0[/tex]
- Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}(1)[/tex]
- Simplify: [tex]\displaystyle \int\limits^{\frac{\pi}{24}}_0 {\cos (12x)} \, dx = \frac{1}{12}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
