Question

plzz solve it i m not getting my answer​

Answer 1

EXPLANATION.

$\sf \: \implies \: \int \: \dfrac{ {x}^{2} + 1}{ {x}^{4} - {x}^{2} + 1 }dx \\ \\ \sf \: \implies \: divide \: numerator \: \: and \: \: denominator \: by \: {x}^{2} \\ \\ \sf \: \implies \: \int \: \frac{ \dfrac{ {x}^{2} }{ {x}^{2} } + \dfrac{1}{ {x}^{2} } }{ \dfrac{ {x}^{4} }{ {x}^{2} } - \dfrac{ {x}^{2} }{ {x}^{2} } + \dfrac{1}{ {x}^{2} } } dx \\ \\ \\ \sf \: \implies \: \int \: \frac{1 + \dfrac{1}{ {x}^{2} } }{ {x}^{2} - 1 + \dfrac{1}{ {x}^{2} } } dx$

$\sf \: \implies \: replace \: = - 1 = - 2 + 1 \\ \\ \sf \: \implies \: \int \: \frac{1 + \dfrac{1}{ {x}^{2} } }{ {x}^{2} + \frac{1}{ {x}^{2} } - 2 + 1} dx \\ \\ \sf \: \implies \: \int \: \dfrac{1 + \dfrac{1}{ {x}^{2} } }{(x - \dfrac{1}{x}) {}^{2} + 1 } dx$

$\sf \: \implies \: let \: u \: = \: x - \dfrac{1}{x} \\ \\ \sf \: \implies \: differentiate \: w.r.t \: \: to \: \: u \\ \\ \sf \: \implies \: du \: = (1 + \frac{1}{ {x}^{2} } )dx \\ \\ \sf \: \implies \: \int \: \frac{du}{ {u}^{2} + 1 }$

$\sf \: \implies \: \tan {}^{ - 1} ( \dfrac{u}{1} ) + c \\ \\ \sf \: \implies \: \tan {}^{ - 1} (x - \frac{1}{ {x}^{} } ) + c \\ \\ \sf \: \implies \: \tan {}^{ - 1} ( \frac{ {x}^{2} - 1 }{x} ) + c$

Option C is correct.