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Evaluate: int(1+x^2)/((1/x-x)sqrt(1+x^2...

Evaluate: `int(1+x^2)/((1/x-x)sqrt(1+x^2+x^4))dx`

Text Solution

Verified by Experts

We have,
`int(1+x^(2))/(1-x^(2))*(1)/(sqrt(1+x^(2)+x^(4))dx=int((1)/(x)+x)/(((1)/(x)-x)xsqrt(x^(2)+(1)/(x^(2))+1))dx`
`=-int(1+(1)/(x^(2)))/(x-(1)/(x))*(1)/(sqrt((x-(1)/(x))^(2)+3))dx`
`=-int(d(x-(1)/(x)))/((x-(1)/(x))sqrt((x-(1)/(x))^(2)+3))dx`
`=-int(du)/(usqrt(u^(2)+3))`, `u=x-(1)/(x)` (say)
`=-int(-(1)/(z^(2))dz)/((1)/(z)sqrt((1)/(z^(2))+3))=int(1)/(sqrt(1+3z^(2)))dz`, `z=(1)/(u)` (say)
`=(1)/(sqrt(3))int(1)/(sqrt(z^(2)+((1)/(sqrt(3)))^(2)))dz`
`=(1)/(sqrt(3))log|z+sqrt(z^(2)+(1)/(3))|+C`
`=(1)/(sqrt(3))log|(x)/(x^(2)-1)+sqrt(((x)/(x^(2)-1))^(2)+(1)/(3))|+C`
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