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Question Number 138580 by Bekzod Jumayev last updated on 15/Apr/21

Commented by Bekzod Jumayev last updated on 15/Apr/21

$$Pleasehelp?$$$$$$

Answered by Ar Brandon last updated on 15/Apr/21

$${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x})\mathrm{dx}$$$${\int }_{-\pi }^{\pi }\frac{\sin \left(2021\mathrm{x}\right)}{(1+3^{\mathrm{x}})\mathrm{sinx}}\mathrm{dx}={\int }_{-\pi }^{\pi }\frac{\sin \left(2021\mathrm{x}\right)}{(1+3^{-\mathrm{x}})\mathrm{sinx}}\mathrm{dx}={\int }_{-\pi }^{\pi }\frac{3^{\mathrm{x}}\sin \left(2021\mathrm{x}\right)}{(1+3^{\mathrm{x}})\mathrm{sinx}}\mathrm{dx}$$$$=\frac{1}{2}{\int }_{-\pi }^{\pi }\frac{(1+3^{\mathrm{x}})\sin \left(2021\mathrm{x}\right)}{(1+3^{\mathrm{x}})\mathrm{sinx}}\mathrm{dx}=\frac{1}{2}{\int }_{-\pi }^{\pi }\frac{\sin \left(2021\mathrm{x}\right)}{\mathrm{sinx}}\mathrm{dx}$$$$={\int }_0^{\pi }\frac{\sin \left(2021\mathrm{x}\right)}{\mathrm{sinx}}\mathrm{dx}$$

Answered by phanphuoc last updated on 15/Apr/21

$$I={\int }_{-\pi }^{\pi }{3}^{x}sin\left(2021x\right)dx/(1+3^x)sinxwithx=-u$$$$2I={\int }_{-\pi }^{\pi }sin\left(2021x\right)dx/sinx$$

Answered by mnjuly1970 last updated on 15/Apr/21

$${\mathit{\varphi }}_n={\int }_0^{\pi }\frac{sin\left(nx\right)}{sin\left(x\right)}........(\ast )$$$${\mathit{\varphi }}_{n-2}={\int }_0^{\pi }\frac{sin\left((n-2)x\right)}{sin\left(x\right)}dx$$$${\mathit{\varphi }}_n-{\mathit{\varphi }}_{n-2}={\int }_0^{\pi }\frac{2cos\left(\frac{nx+nx-2x}{2}\right)sin\left(\frac{2x}{2}\right)}{sin\left(x\right)}dx$$$$={\int }_0^{\pi }\frac{cos\left((n-1)x\right).sin\left(x\right)}{sin\left(x\right)}dx$$$$=\frac{1}{n-1}{\int }_0^{\pi }cos\left((n-1)x\right)dx=0$$$${\mathit{\varphi }}_n={\mathit{\varphi }}_{n-2}......✓$$$$(\ast )......::{\mathit{\varphi }}_0=0,{\mathit{\varphi }}_1=\pi ,{\mathit{\varphi }}_2=0\left(✓\right)..$$$${\mathit{\varphi }}_n=\{\begin{array}{l}0if;n:=even\\ \pi if;n:=odd\end{array}$$$$..........\therefore {\mathit{\varphi }}_{2021}=\pi ..........$$$$$$

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