Question

Please help with this question from indefinite integrals.
Also, tell me if you think there is something wrong with this question because I think so.

Answer 1

Differentiate above equation on both sides with respect to 'x'.
L.H.S : $\frac{( x^{2} - 1) }{( x^{4}+3 x^{2} +1)( Tan^{-1}(x + 1/x)) }$
R.H.S :
Let $Tan^{-1} (x + 1/x)$ = f(x)
Then differentiating log(f(x)) w.r.t 'x' is [1/f(x)]$f^{'}(x)$
Here, $f^{'}(x)$ = $\frac{d( tan^{-1}(x + 1/x) }{x}$
    we know that, $\frac{d( tan^{-1} t)}{t}$ = 1/(1 + $t^{2}$)
replace 't' with (1 + 1/x) and on further simplification, you will get LHS.