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The integral int(2)^(4)(logx^(2))/(logx^...

The integral `int_(2)^(4)(logx^(2))/(logx^(2)+log(36-12x+x^(2))) dx` is equal to

A

2

B

4

C

1

D

6

Text Solution

Verified by Experts

The correct Answer is:
C
PLAN Apply the property
`int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx` and then add. ltbtgt Let `I=int_(2)^(4)(logx^(2))/(logx^(2)+log(36-12x+x^(2)))dx`
`=int_(2)^(4)(2logx)/(2 logx+log(6-x)^(2))dx`
`=int_(2)^(4)(2logxdx)/(2[logx+log(6-x)]]`
`rArrI=int_(2)^(4)(log x dx)/([logx+log(6-x)]]dx` . . . (i)
`rArrI=int_(2)^(4)(log(6-x))/(log(6-x)+logx)dx` . . . (ii)
`[:'int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx]`
On adding Eqs . (i) and (ii) , we get
`2I=int_(2)^(4)(logx+(6-x))/(logx+log(6-x))dx`
`rArr2I=int_(2)^(4)dx=[x]_(2)^(4)rArr2I=2`
`rArr2I=2rArrI=1`
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