Evaluate: (i) `intx/(sqrt(x^2+a^2)+sqrt(x^2-a^2))\ dx` (ii) `int(xtan^(-1)x^2)/(1+x^4)\ dx`

Explanation

(i)
`int frac{x d x}{sqrt{x^{2}+a^{2}}+sqrt{x^{2}-a^{2}}}`

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

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

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

`text { Now, } int frac{x d x}{sqrt{x^{2}+a^{2}}+sqrt{x^{2}-a^{2}}}`

`=frac{1}{2} int frac{d t}{sqrt{t+a^{2}}+sqrt{t-a^{2}}}`

`=frac{1}{2} int frac{d t}{(sqrt{t+a^{2}}+sqrt{t-a^{2}})} times frac{(sqrt{t+a^{2}}-sqrt{t-a^{2}})}{(sqrt{t+a^{2}}-sqrt{t-a^{2}})}`

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