Mathematics - Functions Question with Solution | TestHub

$f^{\prime }(x)-\frac{2x}{x^2+1}f(x)=-\frac{\left(x^2+1\right)}{x^2}\Rightarrow I.F=e^{\int \frac{-2x}{x^2+1}dx}=\frac{1}{x^2+1}$ ∴ solution is $f(x)\times \frac{1}{x^2+1}=\int \frac{-\left(x^2+1\right)}{x^2}\times \frac{1}{x^2+1}dx=\frac{1}{x}+c$ As $f(1)=2\Rightarrow c=0\Rightarrow f(x)=x+\frac{1}{x}$ $g^{\prime }(x)=\frac{2(x+2)(x-2)(x+1)(x-1)\left(x^2+1\right)}{x^3\left(1+x^4\right)}$