Question

please pzz Solve
this question only a genius can solve it....

it's answer is----

$\tan( { }^{ - 1} ) ((x {}^{2} - 1) \div x)$

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Answer 1

$\underline{\text{Solution :}}$

$\mathrm{Now,\:\int \frac{x^{2}+1}{x^{4}-x^{2}+1}dx}$

$\mathrm{=\int \frac{x^{2}+1}{(x^{2}-1)^{2}+x^{2}}dx}$

$\mathrm{=\int \frac{1+\frac{1}{x^{2}}}{(\frac{x^{2}-1}{x})^{2}+1}dx}$

$\mathrm{=\int \frac{d(\frac{x^{2}-1}{x})}{(\frac{x^{2}-1}{x})^{2}+1}}$

$\mathrm{=tan^{-1}\frac{x^{2}-1}{x}+C}$

$\text{where C is integral constant}$

$\to \mathrm{\int \frac{x^{2}+1}{x^{4}-x^{2}+1}dx = tan^{-1}\frac{x^{2}-1}{x}+C}$