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Evaluate int(1+x^(4))/((1-x^(4))^(3//2))...

Evaluate `int(1+x^(4))/((1-x^(4))^(3//2))dx.`

Text Solution

Verified by Experts

`I=int(1+x^(4))/((1-x^(4))^(3//2))dx=int(x^(3)(x+1//x^(3))dx)/(x^(3)(x^(2)-(1)/(x^(2)))^(3//2))`
`=int((x+1//x^(3))dx)/(((1)/(x^(2))-x^(2))^(3//2))`
Let `(1)/(x^(2))-x^(2)=t or ((-2)/(x^(3))-2x)dx=dt`
or `(x+(1)/(x^(3)))dx= -(1)/(2)dt`
` :. I= -(1)/(2)int(dt)/(t^(3//2))=(1)/(sqrt(t))+C=(1)/(sqrt((1)/(x^(2))-x^(2)))+C`
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