Evaluate: `int1/(x(x^6+1))dx`

Explanation

`int frac{d x}{x(x^{6}+1)}`

`=int frac{x^{5} d x}{x^{6}(x^{6}+1)}`

`text { let } x^{6}=t`

`Rightarrow 6 x^{5} d x=d t`

`Rightarrow x^{5} d x=frac{d t}{6}`

`text { Now, } int frac{d x}{x^{6}(x^{6}+1)}`

`=frac{1}{6} int frac{d t}{t(t+1)}`

`=frac{1}{6} int frac{d t}{t^{2}+t}`

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