Question

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

Answer 1

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

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^{(1)/(4)}_{(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 = (1)/(4) ,\ u = 2\pi ((1)/(4)) = (\pi)/(2)} \atop {x = (1)/(12)} ,\ u = 2\pi ((1)/(12)) = (\pi)/(6)} \right.`

Step 3: integrate Pt. 2

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

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

Unit: Integration