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If f(x) = int(5x^(8)+7x^(6))/((x^(2)+1+2...

If `f(x) = int(5x^(8)+7x^(6))/((x^(2)+1+2x^(7))^(2))dx, (x ge 0)`, and f(0) = 0, then the value of f(1) is

A

`-1//2`

B

`1//4`

C

`1//2`

D

`-1//4`

Text Solution

Verified by Experts

The correct Answer is:
B
`f(x)=int(5x^(8)+7x^(6))/(x^(14)(2+(1)/(x^(7))+(1)/(x^(5)))^(2))dx`
`=int((5)/(x^(6))+(7)/(x^(8)))/((2+(1)/(x^(7))+(1)/(x^(5)))^(2))dx`
`"Put, "2+(1)/(x^(7))+(1)/(x^(5))=t`
`rArr" "f(x)=-int(dt)/(t^(2))`
`=(1)/(t)+c`
`=((x^(7))/(2x^(7)+x^(2)+1))+c`
`" "f(0)=0`
`rArr" "c=0`
`rArr" "f(x)=((x^(7))/(2x^(7)+x^(2)+1))`
`rArr" "f(1)=(1)/(4)`
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