Mathematics

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


SOLUTION

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

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Subjective Medium Published on 17th 09, 2020
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