jee-advanced 2009 Q24

jee-advanced · India · paper2 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution

If $$I_{n}=\int_{-\pi}^{\pi}\frac{\sin nx}{\left(1+\pi^{x}\right)\sin x}dx,\quad n=0,1,2,\ldots,$$ then
(A) $I_{n}=I_{n+2}$
(B) $\sum_{m=1}^{10}I_{2m+1}=10\pi$
(C) $\sum_{m=1}^{10}I_{2m}=0$
(D) $I_{n}=I_{n+1}$

(A), (B), (C)

If
$$I_{n}=\int_{-\pi}^{\pi}\frac{\sin nx}{\left(1+\pi^{x}\right)\sin x}dx,\quad n=0,1,2,\ldots,$$
then\\
(A) $I_{n}=I_{n+2}$\\
(B) $\sum_{m=1}^{10}I_{2m+1}=10\pi$\\
(C) $\sum_{m=1}^{10}I_{2m}=0$\\
(D) $I_{n}=I_{n+1}$

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