If ${I_n} = \mathop \smallint \limits_0^{\pi /2} \,\,\frac{{{{\sin }^2}\,\,n\,\,{\rm{ }}x}}{{{{\sin }^2}\,\,x}}\,\,dx$, then which one of the following is not true?

${I_n} = 2\mathop \smallint \limits_0^{\pi /2} \,\,\frac{{\sin \,\,x\,\,\cos \,\,2n{\rm{ }}\,x}}{{\sin \,\,x}}\,\,dx$

${I_1},\,\,{I_2},\,\,{I_3},\,\, \ldots {I_n} \ldots .$ Is a A.P.

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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{I_n} = \int \limits_0^{\pi /2} \,\,\frac{{{{\sin }^2}\,\,n\,\,\mathrm{ }x}}{{{{\sin }^2}\,\,x}}\,\,dx

, then which one of the following is not true?

View answer and explanation

Correct option: D

\sin \,\,\left( {{I_{15}}} \right) = 0

Option D (sin(I_15) = 0) is correct because it is not necessarily true, as I_n = n*pi/2 and sin(15*pi/2) is not 0.

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