Mathematics

Evaluate :$$\displaystyle \int \dfrac{dx}{x(x^2+1)}$$


SOLUTION
$$ \displaystyle I = \int \dfrac{x}{x^{2}(x^{2}+1)}dx $$ 
let $$ x^{2} = t \Rightarrow 2xdx = dt $$
$$ \Rightarrow I = \displaystyle \int \frac{dt}{2t(t+1)}(t+1-t ) = \frac{1}{2}\left [ \frac{dt}{t}-\dfrac{dt}{t+1} \right ]$$
$$ = \dfrac{1}{2} (lnt - ln|t+1|) - \dfrac{1}{2}(ln(x^{2})-ln(1+x^{2}))+c$$
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Subjective Medium Published on 17th 09, 2020
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