Question

plz answer this question guys...​

Answer 1

Given :

$\displaystyle{\sf \int \dfrac{x^2+1}{x^4+1} \:dx}$

Solution :

$\displaystyle{\sf \implies \int \dfrac{x^2+1}{x^4+1} \:dx}$

$\displaystyle{\sf \implies \int \dfrac{\Big(1+\dfrac{1}{x^2}\Big)}{\Big(x^2+\dfrac{1}{x^2}\Big)} \:dx}$

$\displaystyle{\sf \implies \int \dfrac{1+\dfrac{1}{x^2}}{2-2+x^2+\dfrac{1}{x^2}}}$

Using Substitution,

$\boxed{\sf x-\dfrac{1}{x}=t \Rightarrow 1+\dfrac{1}{x^2}\:dx=dt}$

$\displaystyle{\sf \implies \int \dfrac{1}{t^2+2}\:dt}$

$\displaystyle{\sf \implies \dfrac{1}{\sqrt{2}}\:tan^{-1}\Big(\dfrac{t}{\sqrt{2}}\Big)+C}$

$\displaystyle{\implies \sf \dfrac{1}{\sqrt{2}} \:tan^{-1}\bigg(\dfrac{x^2-1}{\sqrt{2}x}\bigg)+C}$

Required Answer :

$\boxed{\boxed{\displaystyle{\sf \int \dfrac{x^2+1}{x^4+1}\:dx = \dfrac{1}{\sqrt{2}}\:tan^{-1}\bigg(\dfrac{x^2-1}{\sqrt{2}x}\bigg) +C}}}$