Evalute : 1 x (x^5+1 ) d x - Vidyadip

Question

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

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Answer

$
\begin{aligned}
& \text { Let } I =\int \frac{1}{x\left(x^5+1\right)} d x \\
& =\int \frac{x^4}{x^5\left(x^5+1\right)} d x
\end{aligned}
$
Put $x^5=t$. Then $5 x^4 d x=d t$
$
\begin{array}{rl}
\therefore x^4 & d x=\frac{d t}{5} \\
\therefore I & =\int \frac{1}{t(t+1)} \cdot \frac{d t}{5} \\
& =\frac{1}{5} \int \frac{(t+1)-t}{t(t+1)} d t \\
& =\frac{1}{5} \int\left(\frac{1}{t}-\frac{1}{t+1}\right) d t \\
& =\frac{1}{5}\left[\int \frac{1}{t} d t-\int \frac{1}{t+1} d t\right] \\
& =\frac{1}{5}[\log |t|-\log |t+1|]+c
\end{array}
$
$
=\frac{1}{5} \log \left|\frac{t}{t+1}\right|+c=\frac{1}{5} \log \left|\frac{x^5}{x^5+1}\right|+c \text {. }
$

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