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Integrate the rational functions1/(x(x^...

Integrate the rational functions`1/(x(x^n+1)`[Hint: multiply numerator and denominator by `x^(n-1)`and put `x^n=t`]

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`" Let " I= int(1)/(x(x^(n)+1))dx= int(x^(n-1))/(x^(n)(x^(n)+1))dx`
`:. I= int(dt)/(n.t(t+1))" "underset(rArr n.x^(n-1)dx-dt)(" Let " x^(n)=t)`
`=(1)/(n)int((t+1)-t)/(t(t+1))dt" " rArr x^(n-1) dx=(dt)/(n)`
` =(1)/(n) int((1)/(t)-(1)/(t+1))dt`
`=(1)/(n) [log |t|-log (t+1)]+c`
`=(1)/(n)log |(t)/(t+1)|+C`
`=(1)/(n)log |(x^(n))/(x^(n)+1)|+C`
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