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Physics Helpline

L K Satapathy

Definite Integrals 7

© Physics

Helpline

Physics Helpline

L K Satapathy

Definite Integrals - 7

Question : is equal to1

2

( 1)( 2). . . 3limn

nn

n n nn

4 2 2

18 27 9( ) ( ) ( ) ( ) 3log3 2a b c de e e

Answer : 1

2

( 1)( 2). . . 3limn

nn

n n nLet Pn

1

2

( 1)( 2). . . ( 2 )limn

nn

n n n nn

1( 1) ( 2) ( 2 )lim . . . .n

n

n n n nn n n

11 2 2lim 1 . 1 . . . 1

n

n

nn n n

© Physics

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Physics Helpline

L K Satapathy

Definite Integrals - 7

11 2 2log lim log 1 . 1 . . . 1

n

n

nPn n n

We use

1 1 2 2lim log 1 . 1 . . . 1n

nn n n n

2

1

1lim log 1n

n r

rn n

1 1 2 2lim log 1 log 1 . . . log 1n

nn n n n

22

1 0

1log lim log 1 log(1 ).n

n r

rP x dxn n

1 0r x

2 2

0 1 1

( ) lim . ( ) lim . ( )2

b n n

h nr ra

b af x dx h f a rh f a rhn

&rx nn

2 2r n x 2

0

b

a

2 0 12 2b an n n

2 2n terms n n

© Physics

Helpline

Physics Helpline

L K Satapathy

Definite Integrals - 7

Integrating by parts : Put

u dv u v vdu

1log(1 ) &1

u x du dv dx v xx

2 2

2

00 0

log(1 ). log(1 ) .1xx dx x x dxx

2

0

1 1(2log3 0) .1x dxx

2 2

0 0

2log31dxdxx

2 2

0 02log3 log(1 ) 2log3 2 log3x x

log 3log3 2 log 27 2P

log 27log 27 2

2 2

2 [7 ]eP ee

A se

n

Correct option = (b)

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Physics Helpline

L K Satapathy

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