If `I_n=int_(-pi)^(pi) \ (sinnx)/((1+pi^x) \ sinx) \ dx, n=0,1,2,......` then which one of the following is not true ?

Options

A. `I_(n)=I_(n+2)`
B. `sum_(m=1)^(10)I_(2m+1)=10pi`
C. `sum_(m=1)^(10)I_(2m)=0`
D. `I_(n)=I_(n+1)`

Answer

A, B, C

Explanation

`I_(n)=int_(-pi)^(pi)(sin nx)/((1+pi^(x))sinx)dx`
`=int_(0)^(pi)((sin nx)/((1+pi^(x))sin x)+(pi^(x)sin nx)/((1+pi^(x))sinx))dx=int_(0)^(pi)(sin nx)/(sin x)dx`
Now `I_(n_2)-I_(n)=int_(0)^(pi)(sin(n+2)x-sin nx)/(sinx)dx`
`=int_(0)^(pi)(2cos(n+1)x sinx)/(sin x)dx=0`
`:. I_(1)=pi,I_(2)=int_(0)^(pi)2cosx dx=0`