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Integral Calculus

Blazing goIITian

Joined:	9 May 2009
Post:	766

9 Nov 2009 16:46:11 IST

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1. integrate: x sin(2x) sin(x/2) cos(x) / (2x - pi)...................limit from 0 to pi

2. Evaluate: lt n-> infinity (1/n) [ ( 1/ n+1) + (2 / n+2) + .......... + (3n/4n) ]

3. If f : R -> R, g : R ->R are continuous functions, then prove that the value of integral

int [ f(x) + f(-x) ] [ g(x) + g(-x) ] dx.........( limit from pi to pi/2)..........is zero.

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kabi

Blazing goIITian

Joined: 11 Jan 2009

Posts: 575

9 Nov 2009 18:13:43 IST

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2 . lim ( n --> infty) (1/n) [ ( 1/ n+1) + (2 / n+2) + .......... + (3n/4n) ]

=lim ( n --> infty) (1/n) (1/(n+1) +1/(n/2 +1)+....................+1/(n/3n+1) ]

= lim ( n --> infty) (1/n) sigma ( r from 1 to 3n ) [ 1/(n/r +1) ]

= integration ( 0 to 3 ) x / (x+1) dx ( put r/n =x )

=x - log (x+1)

after putting limits

=3-log(4)

thank u

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