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Evaluate int(x dx)/(x^(3)sqrt(x^(2)-1))....

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

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Verified by Experts

` " Write " I=int(x dx)/(x^(4)sqrt(x^(2)-1)) " and put " x^(2)-1=t^(2), " so that " `
`2x dx=2t dt`
`:. I=int(t)/((t^(2)+1)^(2)t)dt=int(dt)/((t^(2)+1)^(2))`
` " But " tan^(-1)t=int(dt)/(t^(2)+1)=int1*(1)/(t^(2)+1)dt`
`=(t)/(t^(2)+1)+int t(2t)/((t^(2)+1)^(2))dt`
` =(t)/(t^(2)+1)+2int(t^(2)+1-1)/((t^(2)+1)^(2))dt`
`=(t)/(t^(2)+1)+2tan^(-1)t-2I`
`:. I=(1)/(2)(t)/(t^(2)+1)+(1)/(2) tan^(-1)t +C`
`=(1)/(2)((sqrt(x^(2)-1))/(x^(2))+"tan"^(-1)sqrt(x^(2)-1))+C`
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