Integration Definite

VITEEE 2015 Papers › Algebra questions › Integration Definite

Eva.uate : \int_{-2n}^{2n+\frac 12}{\sin(\pi x)\left\{\frac x2 \right\}} \;\; \mathrm{d}x

#Mathematics #Algebra

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9 Answers

262

Aditya Bhutra ·2012-03-22 06:50:47

is n integer ?

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262

Aditya Bhutra ·2012-03-22 07:23:09

i am getting 12pi² - 2npi

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rishabh ·2012-03-22 09:41:20

im geting exactly same as aditya,
break it from -2n to 2n and 2n to 2n + 1/2.
put {x} = x-[x]
so we hav 3 integrals out of which the one with [x] is zero. 'cos when u break it into many integrals (as we so generally for any integral involving step function) each term on integration will be 0..

now from by parts use âˆ«xsinx = sinx - xcosx for the remaining 2 integrals.

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Vivek @ Born this Way ·2012-03-23 06:57:34

But the answer has no 2 for one term ie., 1Î ² - 2nÎ .

Aditya, How did you do and it? Also one point to note is that both function are periodic with period 2. Hence job is also more easier.

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rishabh ·2012-03-23 07:04:13

do mention if it's your doubt or no. it'll save me time writing the steps.

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262

Aditya Bhutra ·2012-03-23 07:05:57

calculate the integral from 0 to 2 .

now since the function has a period 2,

given integral = 2n * (integral from 0 to 2) + (integral from 0 to 1/2)

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Vivek @ Born this Way ·2012-03-23 07:46:16

Hmm thanks aditya/risabh.

@Rishbah. Yes I wished to know the method pursued by u.

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blueivy ·2012-03-29 11:31:20

I am also getting the same as aditya, simply by parts. 1st is x/2 and second fn is sinpix

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souradipta Sen ·2012-04-21 03:16:49

breaking down the integration
_-2nâˆ«²ⁿsin(Î x){0.5x}dx+_2nâˆ«^2n+0.5sin(Ï€x)sin(Î x){0.5x}dx

first is odd and periodic hence=0 period 2

second 1:
₀âˆ«^0.50.5xsin(Î x)dx

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