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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

= - \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

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

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

Füge eine Konstante zur Lösung hinzu

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

\int sin (3 x) sin (2 x) d x

d er i v a t i v e x^{2} - 8 x + 16

\int_{2}^{\infty} \frac{1}{x ^{3}} d x

y^{^{'}} - 2 y = 3 e^{t}

d er i v a t i v e 15000 + 250 0^{t}