Evaluate: 2x (x^ 2 +1)(x^ 2 +3) dx. - Vidyadip

Question

Evaluate: $\int\frac{\text{2x}}{\text{(x}^{2}+1)\text{(x}^{2}+3)}\text{dx}$.

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Answer

$\text{I}=\int\frac{\text{2x}}{\text{(x}^{2}+1){\text{(x}^{2}+3)}}\text{dx}=\int\frac{\text{dt}}{\text{(t + 1)(t + 3)}}\text{where t = x}^{2}$
$=\frac{1}{2}\int\Bigg[\frac{1}{\text{(t + 1)}}-\frac{1}{{\text{(t + 3)}}}\Bigg]\text{dt}$
$=\frac{1}{2}[\log|\text{t + 1}|-\log|\text{t + 3}|]+\text{c}$
$=\frac{1}{2}[\log\text{(x}^{2}+1)-\log\text{(x}^{2}+3)]+\text{c}$.

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