Question

$I_n={\int }_0^{\frac{\pi }{2}}{\cos }^{n}x\cos \left(nx\right)dx,nϵN$ then $\sqrt{I_{2001}:I_{2002}}$ can be the eccentricity of

• A. Parabola
• B. Ellipse
• C. Circle
• D. Hyperbola

Answer 1

The correct option is D Hyperbola

$I_{n+1}={\int }_0^{\frac{\pi }{2}}{\cos }^{n+1}x\cos (n+1)xdx={\int }_0^{\frac{\pi }{2}}{\cos }^{n+1}x(\cos nx\cos x-\sin nx\sin x)dx\therefore I_{n+1}=I_n-I_{n+1}\Rightarrow 2I_{n+1}=I_n\therefore I_n:I_{n+1}=2$