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Evaluate : int(0)^(1) (x^(2))/(1+x^(6)) ...

Evaluate : `int_(0)^(1) (x^(2))/(1+x^(6)) dx`

Text Solution

Verified by Experts

Let `x^(3) =t`
`rArr " "3x^(2) dx= dt`
`rArr " "x^(2) dx =(dt)/(3)`
`At x=0 " "t=0^(3) =0`
`At x=1, " "t=1^(3) =1`
`:. int_(0)^(1) (x^(2))/(1+x^(6)) dx= int_(0)^(1) (dt)/(3(t+t^(2)))`
` =1/3 [tan^(-1) t]_(0)^(1)`
`=1/3 [tan^(-1) 1-tan^(-1)0]`
`1/3((pi)/(4)-0) =(pi)/(12).`
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