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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 ?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)` |
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Answer» Correct Answer - A::B::C `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` |
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