Question

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

Answer 1

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

This can be written as

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

This again can be written as

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

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

This can be simplified as,

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

$\Rightarrow x-2{\tan }^{-1}x+C$

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