Evaluate the following integrals:


Evaluate $\int \frac{1}{x \sqrt{1+x^{n}}} d x$


Let, $\sqrt{1+x^{n}}=t$

Differentiate both side with respect to $\mathrm{t}$

$\frac{n x^{n-1}}{2 \sqrt{1+x^{n}}} \frac{d x}{d t}=1 \Rightarrow \frac{d x}{x \sqrt{1+x^{n}}}=\frac{2 d t}{n\left(t^{2}-1\right)}$

$y=\int \frac{2}{n\left(t^{2}-1\right)} d t$

Use formula $\int \frac{1}{t^{2}-a^{2}} d t=\frac{1}{2 a} \ln \left(\frac{t-a}{t+a}\right)$

$y=\frac{1}{n} \ln \left(\frac{t-1}{t+1}\right)+c$

Again put $t=\sqrt{1+x^{n}}$

$y=\frac{1}{n} \ln \left(\frac{\sqrt{1+x^{n}}-1}{\sqrt{1+x^{n}}+1}\right)+c$

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