Evaluate: `intx/(x^4+2x^2+3)\ dx`

Explanation

`int frac{x d x}{x^{4}+2 x^{2}+3} `

`text { let } x^{2}=t `

`Rightarrow 2 xdx=d t `

`Rightarrow xdx=frac{d t}{2} `

`text { Now, } int frac{x d x}{x^{4}+2 x^{2}+3} `

`=frac{1}{2} int frac{d t}{t^{2}+2 t+3} d frac{d t}{t^{2}+2 t+1+2} `

`=frac{1}{2} int t frac{1}{2} int frac{(t+1)^{2}+(sqrt{2})^{2}}{=} `

`=frac{1}{2} times frac{1}{sqrt{2}} tan ^{-1}(frac{t+1}{sqrt{2}})+C[because int frac{d x}{x^{2}+a^{2}}=frac{1}{a} tan ^{-1}(frac{x}{a})+C] `

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