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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)/(4)`

D

`(1)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
C
We have, `f(x) = int(5x^(8)+7x^(6))/((x^(2)+1+2x^(7))^(2))dx`
`=int(5((x^(8))/(x^(14)))+7((x^(6))/(x^(14))))/((x^(2)/(x^(7))+(1)/(x^(7))+(2x^(7))/(x^(7)))^(2))dx`
(dividing both numerator and denominator by `X^(14)`)
`=int(5x^(-6)+7x^(-8))/((x^(-5)+x^(-7)+2)^(2))dx`
Let `x^(-5) + x^(-7)+2 = t`
`rArr (-5x^(-6)-7x^(-8))dx = dt`
`rArr (5x^(-6)+7x^(-8))dx = - dt`
`therefore f(x) = int - (dt)/(t^(2))= - intt^(-2)dt`
`= -(t^(-2+1))/(-2+1)+C=-(t^(-1))/(-1)+C=(1)/(t)+C`
`=(1)/(x^(-5)+x^(-7)+2)+C=(x^(7))/(2x^(7)+x^(2)+1)+C`
`therefore f(0)=0`
`therefore 0 = (0)/(0+0+1)+C rArr C = 0`
`therefore f(x)=(x^(7))/(2x^(7)+x^(2)+1)`
`rArr" "f(1)=(1)/(2(1)^(7)+1^(2)+1)=(1)/(4)`
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