integral of (z^2)/((z-1)^3)

解答

\int \frac{z ^{2}}{( z - 1 ) ^{3}} d z

ln ∣ z - 1 ∣ - \frac{2}{z - 1} - \frac{1}{2 ( z - 1 ) ^{2}} + C

求解步骤

\int \frac{z ^{2}}{( z - 1 ) ^{3}} d z

使用换元积分法

= \int \frac{( u + 1 ) ^{2}}{u ^{3}} d u

乘开

\frac{( u + 1 ) ^{2}}{u ^{3}} : \frac{1}{u} + \frac{2}{u ^{2}} + \frac{1}{u ^{3}}

= \int \frac{1}{u} + \frac{2}{u ^{2}} + \frac{1}{u ^{3}} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u} d u + \int \frac{2}{u ^{2}} d u + \int \frac{1}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{2}{u ^{2}} d u = - \frac{2}{u}

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

= ln ∣ u ∣ - \frac{2}{u} - \frac{1}{2 u ^{2}}

u = z - 1

代回

= ln ∣ z - 1 ∣ - \frac{2}{z - 1} - \frac{1}{2 ( z - 1 ) ^{2}}

解答补常数

= ln ∣ z - 1 ∣ - \frac{2}{z - 1} - \frac{1}{2 ( z - 1 ) ^{2}} + C