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受欢迎的代数 >

展开 1/(w(w+i)^2)

解答

展开

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

解答

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

求解步骤

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

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

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

乘开

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

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

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

有理化:

\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 )}

使用分式法则:

\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 )}

消掉

\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 )}

消掉

\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}

乘开

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}

将

\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}

改写成标准复数形式:

\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

流行的例子

展开 (derivative of (9t-6^3))/((27-3t)^2)

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

展开 ((x+h)^2+1)(xln(x)+e^x)

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

化简 2t(x)x^{''}+(t(x)-4)x^'-x

s im pl i f y 2 t (x) x^{^{''}} + (t (x) - 4) x^{^{'}} - x

展开 (3x-e^{3x})/(3x(e^{3x)-1)}

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

展开 (w(x,y(x,z),z)+x^'+y(x,z)z)(x+z)

e x p an d (w (x, y (x, z), z) + x^{^{'}} + y (x, z) z) (x + z)