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Use integration by parts to integrate sin2x between pi and 0

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Answer:

[tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = 0[/tex]

General Formulas and Concepts:

Calculus

Integration

  • Integrals

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^0_{\pi} {\sin (2x)} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

  1. Set u:                                                                                                             [tex]\displaystyle u = 2x[/tex]
  2. [u] Differentiate:                                                                                             [tex]\displaystyle du = 2 \ dx[/tex]
  3. [Bounds] Switch:                                                                                           [tex]\displaystyle \left \{ {{x = 0 ,\ u = 2(0) = 0} \atop {x = \pi ,\ u = 2 \pi}} \right.[/tex]

Step 3: Integrate Pt. 2

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = \frac{1}{2} \int\limits^0_{\pi} {2 \sin (2x)} \, dx[/tex]
  2. [Integral] U-Substitution:                                                                               [tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = \frac{1}{2} \int\limits^0_{2 \pi} {\sin u} \, du[/tex]
  3. Trigonometric Integration:                                                                           [tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = \frac{1}{2}(-\cos u) \bigg| \limits^0_{2 \pi}[/tex]
  4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:          [tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = \frac{1}{2}(0)[/tex]
  5. Simplify:                                                                                                         [tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = 0[/tex]

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

Unit: Integration

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