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"Let " k(x)=int((x^(2)+1)dx)/(root(3)(x^...

`"Let " k(x)=int((x^(2)+1)dx)/(root(3)(x^(3)+3x+6)) " and " k(-1)=(1)/(root(3)(2)). " Then the value of " k(-2) " is "-.`

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The correct Answer is:
2
`k(x)=int((x^(2)+1)dx)/((x^(3)+3x+6)^(1//3))`
`"Put " x^(3)+3x+6=t^(3)`
`"or " 3(x^(2)+1)dx=3t^(2)dt`
`k(x)=int(t^(2)dt)/(t)=(t^(2))/(2)+C=(1)/(2)(x^(3)+3x+6)^(2//3)+C`
`k(-1)=(1)/(2)(2)^(2//3)+C " or " C=0`
` :. k(x)=(1)/(2)(x^(3)+3x+6)^(2//3),f(-2)=(1)/(2)(-8)^(2//3)`
`=(1)/(2)[(-2)^(3)]^(2//3)=2`
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