Question

integrate the following ​

Answer 1

Given to evaluate,

$\displaystyle\longrightarrow I=\int\dfrac{dx}{x(x^5+3)}$

Taking $x^5$ common from $x^5+3$ like $x^5+3=x^5\left(1+3x^{-5}\right),$

$\displaystyle\longrightarrow I=\int\dfrac{dx}{x\cdot x^5\left(1+3x^{-5}\right)}$

$\displaystyle\longrightarrow I=\int\dfrac{dx}{x^6\left(1+3x^{-5}\right)}$

$\displaystyle\longrightarrow I=\int\dfrac{x^{-6}\ dx}{1+3x^{-5}}\quad\quad\dots(1)$

Put,

$\displaystyle\longrightarrow u=1+3x^{-5}$

$\displaystyle\longrightarrow du=-15x^{-6}\ dx$

$\displaystyle\longrightarrow x^{-6}\ dx=-\dfrac{1}{15}\,du$

Then (1) becomes,

$\displaystyle\longrightarrow I=-\dfrac{1}{15}\int\dfrac{du}{u}$

$\displaystyle\longrightarrow I=-\dfrac{1}{15}\log|u|+C$

Undoing substitution $\displaystyle u=1+3x^{-5},$

$\displaystyle\longrightarrow I=-\dfrac{1}{15}\log\left|1+3x^{-5}\right|+C$

$\displaystyle\longrightarrow\underline{\underline{I=\dfrac{1}{15}\log\left|\dfrac{x^5}{x^5+3}\right|+C}}$