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If f(x)=x^(5)+5x-1 then int(5)^(41)(dx)/...

If `f(x)=x^(5)+5x-1` then `int_(5)^(41)(dx)/((f^(-1)(x))^(5)+5f^(-1)(x))` equals

A

`0`

B

`log_(e)3`

C

`log_(e)4`

D

`log_(e)7`

Text Solution

Verified by Experts

The correct Answer is:
D
`f(x)=x^(5)+5x-1`
`f(1)=5impliesf^(-1)(5)=1`
`f(2)=41impliesf^(-1)(41)=2`
Put `f^(-1)(x)=t`
`:. Iint_(5)^(41)(dx)/((f^(-1)(x))^(5)+5f^(-1)(x))`
`=int_(1)^(2)(f'(t))/(t^(5)+5t)dt`
`=int_(1)^(2)(5t^(4)+5)/(t^(5)+5t)dt`
`=[log_(e)(t^(5)+5t)]_(1)^(2)`
`=log_(e)42-log_(3)6=log_(e)7`
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