Question

Evaluate the definite integral. 1/4 csc(2πt) cot(2πt) dt 1/12

Answer 1

Answer:

$\displaystyle \int\limits^{\frac{1}{4}}_{\frac{1}{12}} {\csc (2\pi t) \cot (2\pi t)} \, dt = \frac{1}{2\pi}$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

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

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^{\frac{1}{4}}_{\frac{1}{12}} {\csc (2\pi t) \cot (2\pi t)} \, dt$

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

1. Set u:                                                                                                             $\displaystyle u = 2\pi t$
2. [u] Differentiate [Basic Power Rule, Derivative Properties]:                       $\displaystyle du = 2\pi \ dt$
3. [Bounds] Switch:                                                                                           $\displaystyle \left \{ {{x = \frac{1}{4} ,\ u = 2\pi (\frac{1}{4}) = \frac{\pi}{2}} \atop {x = \frac{1}{12}} ,\ u = 2\pi (\frac{1}{12}) = \frac{\pi}{6}} \right.$

Step 3: integrate Pt. 2

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 $\displaystyle \int\limits^{\frac{1}{4}}_{\frac{1}{12}} {\csc (2\pi t) \cot (2\pi t)} \, dt = \frac{1}{2\pi}\int\limits^{\frac{1}{4}}_{\frac{1}{12}} {2\pi \csc (2\pi t) \cot (2\pi t)} \, dt$
2. [Integral] U-Substitution:                                                                               $\displaystyle \int\limits^{\frac{1}{4}}_{\frac{1}{12}} {\csc (2\pi t) \cot (2\pi t)} \, dt = \frac{1}{2\pi}\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{6}} {\csc (u) \cot (u)} \, du$
3. [Integral] Trigonometric Integration:                                                             $\displaystyle \int\limits^{\frac{1}{4}}_{\frac{1}{12}} {\csc (2\pi t) \cot (2\pi t)} \, dt = \frac{1}{2\pi}[-\csc (u)] \bigg| \limits^{\frac{\pi}{2}}_{\frac{\pi}{6}}$
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           $\displaystyle \int\limits^{\frac{1}{4}}_{\frac{1}{12}} {\csc (2\pi t) \cot (2\pi t)} \, dt = \frac{1}{2\pi}(1)$
5. Simplify:                                                                                                         $\displaystyle \int\limits^{\frac{1}{4}}_{\frac{1}{12}} {\csc (2\pi t) \cot (2\pi t)} \, dt = \frac{1}{2\pi}$

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

Unit: Integration