Mathematics

$$\int\frac{dx}{x(x^n+1)}$$


ANSWER

$$\frac{1}{n}$$ $$(log{x^n}$$-$$log(x^n +1))+C$$


SOLUTION

$$\\I=\int (\frac{1}{x(x^n+1)})dx\\I=\int (\frac{1}{x^n(x^n+1)})dx\\let x^n=t then nx^{n-1}dx=dt \\\therefore I=(\frac{1}{n})\int (\frac{1}{t(t+1)})dx\\=(\frac{1}{n})\int ((\frac{1}{t})-(\frac{1}{t+1}))dx\\ =(\frac{1}{n})[logt-log(t+1)]+c\\=(\frac{1}{n})[logx^n-log(x^n+1)]+c$$

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