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Differential Calculus

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9 Mar 2010 15:24:41 IST

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INTEGRAL ISinxI/IxI FROM 0 TO Pi >2/pi(1+1/2+1/3+...+1/n)

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INTEGRAL ISinxI/IxI FROM 0 TO Pi >2/pi(1+1/2+1/3+...+1/n)

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Deepak Aggarwal

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Joined: 9 May 2009

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10 Mar 2010 16:34:39 IST

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Let, I = int |sinx| / |x| limit x from 0 to pi/2

Clearly x is positive in the interval 0 to pi/2 and sinx is potive in 0 to pi/2.

So, I = int sinx / x limit x from 0 to pi/2

Opening the series of sinx here, ( refer : http://www.goiit.com/posts/list/community-shelf-some-special-series-important-for-jee-1009938.htm)

I = int (1 - x²/3! + x⁴/5! - ............ ) limit x from 0 to pi/2

I = x - x³/3.3! + x⁵ / 5.5! - ............ limit 0 to pi/2

I = (pi/2) - (pi/2)³/3.3! + (pi/2)⁵ / 5.5! - ............ + (-1)^2n-1 (pi/2)^2n-1 / (2n-1)(2n-1)!

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