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

Lösung

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

- \frac{2}{x - 1} + ln ∣ x ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

= ln ∣ x - 1 ∣ - \frac{1}{x - 1} + ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1}

Vereinfache

ln ∣ x - 1 ∣ - \frac{1}{x - 1} + ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{1}{x - 1} : - \frac{2}{x - 1} + ln ∣ x ∣

= - \frac{2}{x - 1} + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{x - 1} + ln ∣ x ∣ + C

\int \frac{4 x ^{2}}{( 1 - 2 x ^{3} ) ^{4}} d x

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

\int \frac{4 x + 1}{x ^{2} + 1} d x

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

\int \frac{2 x ^{2} + 4 x + 22}{x ^{2} + 2 x + 10} d x