# Use integration by parts to integrate sin2x between pi and 0

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