​ Tìm n biết I=∫π20sinnxcosxdx=12017(n≠−1) ​

Question

Tìm n biết \(I=\int _{0}^{\frac{\pi }{2} }\sin ^{n} x\cos x{\rm d}x =\frac{1}{2017}  \left(n\ne -1\right).\)

 A. 10. B. 1005. C. 2016. D. 2017.

Answer 1

Chọn C

 Ta có \(I=\int _{0}^{\frac{\pi }{2} }\sin ^{n} x\cos x{\rm d}x =\int _{0}^{\frac{\pi }{2} }\sin ^{n} x{\rm d}\left(\sin x\right) =\frac{\sin ^{n+1} x}{n+1} \left|\begin{array}{l} {\frac{\pi }{2} } \\ {0} \end{array}\right. =\frac{1}{n+1} .\)

 Do đó \(I=\frac{1}{2017} \Leftrightarrow \frac{1}{n+1} =\frac{1}{2017} \Leftrightarrow n=2016.\)