Question

solve : $\int \frac{2+3x^2}{x^2(1+x^2)}dx.$

Answer 1

$\int \frac{2+3x^2}{x^2(1+x^2)}dx$

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

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

$=2\int \frac{dx}{x^2}+\int \frac{dx}{1+x^2}$

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

$=2\frac{x^{-2+1}}{-2+1}+{\tan }^{-1}x+c$ where $c$ is the constant of integration.

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