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Evaluate: Evaluate: intx^(-11)(1+x^4)^(-...

Evaluate: Evaluate: `intx^(-11)(1+x^4)^(-1/2)dx`

Text Solution

Verified by Experts

`I=int (dx)/(x^(11)(1+x^(4))^(1//2))=int (dx)/(x^(11)*x^(2)(1+1//x^(4))^(1//2))`
Let `1+(1)/(x^(4))=t^(2) " or " (-4)/(x^(5))dx=2tdt`
`:. I=int(dx)/(x^(13)(1+1//x^(4))^(1//2 ))`
`=-(1)/(4)int(2tdt)/(x^(8)t)`
`=-(1)/(2)int(t^(2)-1)^(2)dt`
`=-(1)/(2)int(t^(4)-2t^(2)+1)dt`
`= -(1)/(2)[(t^(5))/(5)-(2t^(3))/(3)+t]+C, " where " t=sqrt(1+(1)/(x^(4)))`
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