Help needed in an MCQ

Inππ11+2sin(x/2)(sin(nx/2)sin(x/2))2dx \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) In=In+1n1{ I }_{ n }={ I }_{ n+1 }\forall n\ge 1

B) I0,I1,I2,...,InareinAP{ I }_{ 0 },{ I }_{ 1 },{ I }_{ 2 },{ ...,I }_{ n }\quad are\quad in\quad AP

C) m=09I2m=90π\sum _{ m=0 }^{ 9 }{ { I }_{ 2m }=90\pi }

D) m=010Im=55π\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
3 years, 11 months ago

No vote yet
1 vote

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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 - 3 years, 11 months ago

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First substitute x by -x and add to get rid of the denominator.And then evaluate I0I_0,I1I_1,I2I_2.... which will come out to be 0,π,2π.....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 - 3 years, 11 months ago

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Ooh nice :)

Aditya Kumar - 3 years, 11 months ago

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i think you have written the statement wrong it should be -pi to pi , correct it then only answers can be given @Aditya Kumar if it's -pi to pi then the answer would be B,C,D .....just correct it !

A Former Brilliant Member - 2 years, 10 months ago

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I'm sure that it's one of a, b, c or d.

Mehul Arora - 3 years, 11 months ago

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Thanks for enlightening!

Aditya Kumar - 3 years, 11 months ago

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There is no one :(

Nihar Mahajan - 3 years, 11 months ago

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What????

Aditya Kumar - 3 years, 11 months ago

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