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Answer:[tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = 0[/tex]General Formulas and Concepts:CalculusIntegrationIntegralsIntegration 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-SubstitutionStep-by-step explanation:Step 1: DefineIdentify[tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx[/tex]Step 2: Integrate Pt. 1Identify variables for u-substitution.Set u: [tex]\displaystyle u = 2x[/tex][u] Differentiate: [tex]\displaystyle du = 2 \ dx[/tex][Bounds] Switch: [tex]\displaystyle \left \{ {{x = 0 ,\ u = 2(0) = 0} \atop {x = \pi ,\ u = 2 \pi}} \right.[/tex]Step 3: Integrate Pt. 2[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][Integral] U-Substitution: [tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = \frac{1}{2} \int\limits^0_{2 \pi} {\sin u} \, du[/tex]Trigonometric Integration: [tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = \frac{1}{2}(-\cos u) \bigg| \limits^0_{2 \pi}[/tex]Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^0_{\pi} {\sin (2x)} \, dx = \frac{1}{2}(0)[/tex]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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