integral of (3x^2+1)/(x(x-1)^3)

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

= \int \frac{3 x}{( x - 1 ) ^{3}} + \frac{1}{x ( x - 1 ) ^{3}} d x

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

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

\int \frac{1}{x ( x - 1 ) ^{3}} d x = - ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}

= 3 (- \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}) - ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}

Vereinfache

3 (- \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}) - ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}} : - \frac{2}{x - 1} - \frac{2}{( x - 1 ) ^{2}} - ln ∣ x ∣ + ln ∣ x - 1 ∣

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

Füge eine Konstante zur Lösung hinzu

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

\int x (x + 5)^{4} d x

\int cos^{(3)} (5 x) sin^{- 2} (5 x) d x

\frac{\partial}{\partial x} (e^{x^{5}})

\frac{\partial}{\partial x} (\frac{x}{( 1 - x ) ^{2}})

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