Question

Show that $\int \frac{x^2-2}{(x^4+5x^2+4)\tan \left(\frac{x^2+2}{x}\right)}dx=\log {\tan }^{-1}(x+\frac{2}{x})$.

Answer 1

Let $I=\int \frac{x^2+2}{(x^4+5x^2+4){\tan }^{-1}\left(\frac{x^2+2}{x}\right)}dx$

Substitute ${\tan }^{-1}\left(\frac{x^2+2}{x}\right)=t$

$\Rightarrow \frac{x^2+2}{(x^4+5x^2+4)}dx=dt$

So, $I=\int \frac{1}{t}dt=\log t+c$

$\Rightarrow I=\log \left({\tan }^{-1}\left(\frac{x^2+2}{x}\right)\right)+c$

$\Rightarrow I=\log \left({\tan }^{-1}(x+\frac{2}{x})\right)+c$