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Integrate the following w.r.t.x.: (x^(...

Integrate the following w.r.t.x.:
`(x^(2).tan^(-1)(x^(3)))/(1+x^(6))`

Text Solution

Verified by Experts

`Let I=int (x^(2).tan^(-1)(x^(3)))/(1+x^(6))dx=int tan^(-1)(x^(3)).(x^(2))/(1+x^(6))dx`
`"Put "tan^(-1)(x^(3))=t" "therefore (1)/(1+(x^(3))^(2)).3x^(2)dx=dt`
`therefore (x^(2))/(1+x^(6))dx=(dt)/(3)`
`therefore I=int t.(dt)/(3)=(1)/(3)int t dt`
`=(1)/(3).(t^(2))/(2)+c=(1)/(6)[tan^(-1)(x^3)]^(2)+c`
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