integral of 1/(s^2(s-1)^2)

Lösung

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

2 ln ∣ s ∣ - \frac{1}{s} - 2 ln ∣ s - 1 ∣ - \frac{1}{s - 1} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{s ^{2} ( s - 1 ) ^{2}} : \frac{2}{s} + \frac{1}{s ^{2}} - \frac{2}{s - 1} + \frac{1}{( s - 1 ) ^{2}}

= \int \frac{2}{s} + \frac{1}{s ^{2}} - \frac{2}{s - 1} + \frac{1}{( s - 1 ) ^{2}} d s

Wende die Summenregel an:

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

= \int \frac{2}{s} d s + \int \frac{1}{s ^{2}} d s - \int \frac{2}{s - 1} d s + \int \frac{1}{( s - 1 ) ^{2}} d s

\int \frac{2}{s} d s = 2 ln ∣ s ∣

\int \frac{1}{s ^{2}} d s = - \frac{1}{s}

\int \frac{2}{s - 1} d s = 2 ln ∣ s - 1 ∣

\int \frac{1}{( s - 1 ) ^{2}} d s = - \frac{1}{s - 1}

= 2 ln ∣ s ∣ - \frac{1}{s} - 2 ln ∣ s - 1 ∣ - \frac{1}{s - 1}

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ s ∣ - \frac{1}{s} - 2 ln ∣ s - 1 ∣ - \frac{1}{s - 1} + C

\int 6 x^{2} + 1 d x

x \to 0 + lim (- \frac{1}{x ^{2}})

y^{^{''}} - 5 y^{^{'}} + 9 y = x e^{x}

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

\int tan^{5} (x) d x