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Integral Calculus

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13 Mar 2010 08:30:51 IST

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prove that... integration x / (1-cos a sinx) wrt x within limits 0 to pie = pie(pie-a

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prove that... integration x / (1-cos a sinx) wrt x within limits 0 to pie = pie(pie-a)/sin a plzzz reply

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Deepak Aggarwal

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Joined: 9 May 2009

Posts: 766

13 Mar 2010 11:52:33 IST

Like 2 people liked this

I got a different answer to it.

I am getting I = (pi)² / 2 sin(a).

nyhow approach is like dis.

First apply property int x from 0 to a [ f(x) ] = int x from 0 to a [ f(a - x) ] and then apply

int x from 0 to 2a [ f(x) ] = 2 int x from 0 to a [ f(x) ] , if f(2a - x) = f(x)

U'll get,

I = (pi) int x from 0 to (pi/2) ( 1 / 1 - cos(a) sin(x) ).dx

Let I₁ = int x from 0 to (pi/2) ( 1 / 1 - cos(a) sin(x) ).dx

Now put cos(a) sin(x) = y and u'll get

I₁ = int y from 0 to cos(a) ( 1 / (1 - y) (cos²a - y²)^1/2 ).dy

Now put cos²a - y² = 1/t

U will now get, I₁ = (1/sina) int t from 1 to (1 / 1 - cos(a)) [ (cos(a)/sin²(a))² - (t - (1/sin²a))² ]^1/2

So, finally I₁ = pi / 2 sin(a)

So, I = (pi)² / 2 sin(a) [Ans]..............

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