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Help needed in an MCQ

\[ \large \displaystyle I_n \int_{\pi}^\pi \frac1{1 + 2^{\sin(x/2)} } \left( \frac{\sin(nx/2)}{\sin(x/2)} \right)^2 \, dx \]

Hi brilliant! I've a doubt in calculus! The question is above.

The options are:

A) \({ I }_{ n }={ I }_{ n+1 }\forall n\ge 1\)

B) \({ I }_{ 0 },{ I }_{ 1 },{ I }_{ 2 },{ ...,I }_{ n }\quad are\quad in\quad AP\)

C) \(\sum _{ m=0 }^{ 9 }{ { I }_{ 2m }=90\pi } \)

D) \(\sum _{ m=0 }^{ 10 }{ { I }_{ m }=55\pi } \)

More than one options are right

  • Please provide detail solutions!

  • Please don't fluke answers.

Note by Aditya Kumar
10 months, 1 week ago

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First substitute x by -x and add to get rid of the denominator. Then use the steps in the solution of this problem. This should help you. Sudeep Salgia · 10 months, 1 week ago

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First substitute x by -x and add to get rid of the denominator.And then evaluate \(I_0\),\(I_1\),\(I_2\).... which will come out to be \(0,\pi ,2\pi.....\) hence they will be in AP .So B,C,D.

And that problem by Sudeep can also be evaluated like that. Gautam Sharma · 10 months, 1 week ago

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@Gautam Sharma Ooh nice :) Aditya Kumar · 10 months ago

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@Pi Han Goh @Kartik Sharma @Surya Prakash @Sudeep Salgia please help! Aditya Kumar · 10 months, 1 week ago

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I'm sure that it's one of a, b, c or d. Mehul Arora · 10 months, 1 week ago

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@Mehul Arora Thanks for enlightening! Aditya Kumar · 10 months, 1 week ago

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@Mehul Arora There is no one :( Nihar Mahajan · 10 months, 1 week ago

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@Nihar Mahajan What???? Aditya Kumar · 10 months, 1 week ago

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