Evaluate the following integrals:
Question:

Evaluate the following integrals:

$\int \frac{1}{x\left(x^{6}+1\right)} d x$

Solution:

let I $=\int \frac{1}{x\left(x^{6}+1\right)} d x$

$=\int \frac{x^{5}}{x^{6}\left(x^{6}+1\right)} d x$

Let $x^{6}=t \ldots$ (i)

$\Rightarrow 6 \mathrm{x}^{5} \mathrm{dx}=\mathrm{dt}$

$I=\frac{1}{6} \int \frac{1}{t(t+1)} d t$

$I=\frac{1}{6} \int\left(\frac{1}{t}-\frac{1}{t+1}\right) d t$

$I=\frac{1}{6}\left(\int \frac{1}{t} d t-\int \frac{1}{(t+1)} d t\right)$

$I=\frac{1}{6}(\log t-\log (t+1))+c$

$I=\frac{1}{6}\left(\log x^{6}-\log \left(x^{6}+1\right)\right)+c[$ using $(i)]$

$I=\frac{1}{6} \log \frac{x^{6}}{x^{6}+1}+c\left[\log m-\log n=\log \frac{m}{n}\right]$

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