How do you integrate $$\dfrac{1}{(2+x^2)\cdot\sqrt{(1+x^2)}}$$ without making any substitution? I don't know why I dislike substitution.
2026-03-25 15:45:26.1774453526
Integration without making any substitution
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You can get a solution without substitutions if you'll prove that $$\left(\frac{1}{\sqrt2}\operatorname{artanh}\frac{x}{\sqrt{2(1+x^2)}}\right)'=\frac{1}{(2+x^2)\sqrt{1+x^2}}.$$