What is expand 1/(w(w+i)^2) ?

The solution to expand 1/(w(w+i)^2) is (-1+w^2)/(w(1+w^2)^2)-i 2/(1+w^4+2w^2)

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expand 1/(w(w+i)^2)

expand

\frac{1}{w ( w + i ) ^{2}}

\frac{- 1 + w ^{2}}{w ( 1 + w ^{2} ) ^{2}} - i \frac{2}{1 + w ^{4} + 2 w ^{2}}

Solution steps

\frac{1}{w ( w + i ) ^{2}}

(w + i)^{2} = w^{2} + 2 i w - 1

= \frac{1}{w ( w ^{2} + 2 i w - 1 )}

Expand

w (w^{2} + 2 i w - 1) : w^{3} + 2 i w^{2} - w

= \frac{1}{w ^{3} + 2 i w ^{2} - w}

Rationalize

\frac{1}{w ^{3} + 2 i w ^{2} - w} : \frac{w ^{2} - 2 i w - 1}{w ( w ^{4} + 2 w ^{2} + 1 )}

= \frac{w ^{2} - 1 - 2 i w}{w ( w ^{4} + 2 w ^{2} + 1 )}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{w ^{2} - 1 - 2 i w}{w ( w ^{4} + 2 w ^{2} + 1 )} = \frac{w ^{2}}{w ( w ^{4} + 2 w ^{2} + 1 )} - \frac{1}{w ( w ^{4} + 2 w ^{2} + 1 )} - \frac{2 i w}{w ( w ^{4} + 2 w ^{2} + 1 )}

= \frac{w ^{2}}{w ( w ^{4} + 2 w ^{2} + 1 )} - \frac{1}{w ( w ^{4} + 2 w ^{2} + 1 )} - \frac{2 i w}{w ( w ^{4} + 2 w ^{2} + 1 )}

Cancel

\frac{w ^{2}}{w ( w ^{4} + 2 w ^{2} + 1 )} : \frac{w}{w ^{4} + 2 w ^{2} + 1}

= \frac{w}{w ^{4} + 2 w ^{2} + 1} - \frac{1}{w ( w ^{4} + 2 w ^{2} + 1 )} - \frac{2 i w}{w ( w ^{4} + 2 w ^{2} + 1 )}

Cancel

\frac{2 i w}{w ( w ^{4} + 2 w ^{2} + 1 )} : \frac{2 i}{w ^{4} + 2 w ^{2} + 1}

= \frac{w}{w ^{4} + 2 w ^{2} + 1} - \frac{1}{w ( w ^{4} + 2 w ^{2} + 1 )} - \frac{2 i}{w ^{4} + 2 w ^{2} + 1}

Expand

w (w^{4} + 2 w^{2} + 1) : w^{5} + 2 w^{3} + w

= \frac{w}{w ^{4} + 2 w ^{2} + 1} - \frac{1}{w ^{5} + 2 w ^{3} + w} - \frac{2 i}{w ^{4} + 2 w ^{2} + 1}

Rewrite

\frac{w}{w ^{4} + 2 w ^{2} + 1} - \frac{1}{w ^{5} + 2 w ^{3} + w} - \frac{2 i}{w ^{4} + 2 w ^{2} + 1}

in standard complex form:

\frac{w ^{2} - 1}{w ( w ^{2} + 1 ) ^{2}} - \frac{2}{w ^{4} + 2 w ^{2} + 1} i

= \frac{w ^{2} - 1}{w ( w ^{2} + 1 ) ^{2}} - \frac{2}{w ^{4} + 2 w ^{2} + 1} i

\frac{\frac{d}{d x} ( ( 9 t - 6 ) ^{3} )}{( 27 - 3 t ) ^{2}}

((x + h)^{2} + 1) (x ln (x) + e^{x})

2 t (x) x^{^{''}} + (t (x) - 4) x^{^{'}} - x

\frac{3 x - e ^{3 x}}{3 x ( e ^{3 x} - 1 )}

(w (x, y (x, z), z) + x^{^{'}} + y (x, z) z) (x + z)

What is expand 1/(w(w+i)^2) ?
The solution to expand 1/(w(w+i)^2) is (-1+w^2)/(w(1+w^2)^2)-i 2/(1+w^4+2w^2)