Question

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

Answer 1

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

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

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

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

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

$\Rightarrow -x^{-1}+{\tan }^{-1}x+C$

$\Rightarrow -\frac{1}{x}+{\tan }^{-1}x+C$

Therefore, $\Rightarrow \int \frac{1+2x^2}{x^2(1+x^2)}dx=-\frac{1}{x}+{\tan }^{-1}x+C$