Question

$\huge\tt\bold\green{Question:-}$
Find$\rm{{L}^{-1}\left \{\frac{s}{2s^{2}-1}\right\}$.

Answer 1

$\huge\tt\bold\green{Answer:-}$

SoluTion:- (By Factorization)

$\implies\l\:-\frac{s}{2s^2-1}$

To write  l  as a fraction with a common denominator, multiply by,

$\implies\:\frac{2s^2-1}{2s^2-1}$

$\implies\:l\:\frac{(2s^2-1)}{2s^2-1}\:-\:\frac{s}{2s^2-1}$

Combine the numerators over the common denominator,

$\implies\:\frac{l\:(2s^2-1)\:-s}{2s^2-1}$

Rewrite it in a factored form,

$\implies\:\frac{2ls^2\:-\:l\:-\:s}{2s^2-1}$

$Hope\:It\:Helps\:You\:Mate\::)$

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Answer 2

$\huge\tt\bold\pink{Question:-}$

$\frac{l {}^{ - 1s} }{2} \times s {}^{2} - 1$

$\huge\tt\bold\pink{Answer :-}$

$\large\boxed{\textsf{\textbf{\red{combine \: multiplied\:terms \:into\: a\: simple \: fraction :-}}}}$

$\frac{l {}^{ - 1s} }{2} \times s {}^{2} - 1$

$\frac{l {}^{ - 1ss} }{2} - 1$

$\large\boxed{\textsf{\textbf{\red{combine \: exponents \: :-}}}}$

$\frac{l {}^{ - 1ss} }{3} - 1$

$\large\boxed{\textsf{\textbf{\red{ find \: common \: factor :-}}}}$

$\frac{l {}^{ - 1}s {}^{3} }{2} + \frac{2( - 1)}{2}$

$\large\boxed{\textsf{\textbf{\red{combine \: fractions\: with\:common \: denominator :-}}}}$

$\frac{ {l}^{ - 1}s {}^{3} + 2( - 1)}{2}$

$\large\boxed{\textsf{\textbf{\red{Multiply \: the \: numbers :-}}}}$

$\frac{ {l}^{ - 1}s {}^{3} - 2 }{2}$

$\huge\tt\bold\pink{solution :-}$

$\frac{ {l}^{ - 1}s {}^{3} - 2 }{2}$