integral of (3x^3-8x^2+10)/(x(x-1)^3)

Lösung

\int \frac{3 x ^{3} - 8 x ^{2} + 10}{x ( x - 1 ) ^{3}} d x

13 ln ∣ x - 1 ∣ + \frac{12}{x - 1} - \frac{3}{2 ( x - 1 ) ^{2}} - \frac{1}{( x - 1 ) ^{2}} - 10 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{3 x ^{3} - 8 x ^{2} + 10}{x ( x - 1 ) ^{3}} d x

Multipliziere aus

\frac{3 x ^{3} - 8 x ^{2} + 10}{x ( x - 1 ) ^{3}} : \frac{3 x ^{2}}{( x - 1 ) ^{3}} - \frac{8 x}{( x - 1 ) ^{3}} + \frac{10}{x ( x - 1 ) ^{3}}

= \int \frac{3 x ^{2}}{( x - 1 ) ^{3}} - \frac{8 x}{( x - 1 ) ^{3}} + \frac{10}{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 ^{2}}{( x - 1 ) ^{3}} d x - \int \frac{8 x}{( x - 1 ) ^{3}} d x + \int \frac{10}{x ( x - 1 ) ^{3}} d x

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

\int \frac{8 x}{( x - 1 ) ^{3}} d x = 8 (- \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}})

\int \frac{10}{x ( x - 1 ) ^{3}} d x = 10 (- ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}})

= 3 (ln ∣ x - 1 ∣ - \frac{2}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}) - 8 (- \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}) + 10 (- ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}})

Vereinfache

3 (ln ∣ x - 1 ∣ - \frac{2}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}) - 8 (- \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}) + 10 (- ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}) : 13 ln ∣ x - 1 ∣ + \frac{12}{x - 1} - \frac{3}{2 ( x - 1 ) ^{2}} - \frac{1}{( x - 1 ) ^{2}} - 10 ln ∣ x ∣

= 13 ln ∣ x - 1 ∣ + \frac{12}{x - 1} - \frac{3}{2 ( x - 1 ) ^{2}} - \frac{1}{( x - 1 ) ^{2}} - 10 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= 13 ln ∣ x - 1 ∣ + \frac{12}{x - 1} - \frac{3}{2 ( x - 1 ) ^{2}} - \frac{1}{( x - 1 ) ^{2}} - 10 ln ∣ x ∣ + C

(y^{2} - x y) d x - x^{2} d y = 0

\frac{d}{d x} (a t)

y^{^{'''}} + 14 y^{^{''}} + 49 y^{^{'}} = 0

\int 19 ln (3 x) d x

\int (11 x^{10}) d x