Evaluate the following integrals: x x^4+2x^2+3 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{x}}{\text{x}^4+2\text{x}^2+3}\text{dx}$

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Answer

$\int\frac{\text{x dx}}{\text{x}^4+2\text{x}^2+3}$
Let $\text{x}^2=\text{t}$
$\Rightarrow2\text{x dx = dt}$
$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$
Now, $\int\frac{\text{x dx}}{\text{x}^4+2\text{x}^2+3}$
$=\frac{1}{2}\int\frac{\text{dt}}{\text{t}^2+2\text{t}+3}$
$=\frac{1}{2}\int\frac{\text{dt}}{\text{t}^2+2\text{t}+1+2}$
$=\frac{1}{2}\int\frac{\text{dt}}{(\text{t}+1)^2+(\sqrt2)^2}$
$=\frac{1}{2}\times\frac{1}{\sqrt{2}}\tan^{-1}\Big(\frac{\text{t}+1}{\sqrt{2}}\Big)+\text{C}$ $\Big[\because\int\frac{\text{dx}}{\text{x}^2+\text{a}^2}=\frac{1}{\text{a}}\tan^{-1}\Big(\frac{\text{x}}{\text{a}}\Big)+\text{C}\Big]$
$=\frac{1}{2\sqrt{2}}\tan^{}-1\Big(\frac{\text{x}^2+1}{\sqrt{2}}\Big)+\text{C}$

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