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.

#Calculus

Easy Math Editor

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Comments

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 - 5 years, 6 months ago

@Pi Han Goh @Kartik Sharma @Surya Prakash @Sudeep Salgia please help!

Aditya Kumar - 5 years, 6 months ago

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 - 5 years, 6 months ago

Ooh nice :)

Aditya Kumar - 5 years, 6 months ago

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 - 4 years, 5 months ago

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

Mehul Arora - 5 years, 6 months ago

Thanks for enlightening!

Aditya Kumar - 5 years, 6 months ago

There is no one :(

Nihar Mahajan - 5 years, 6 months ago

What????

Aditya Kumar - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$