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

Lösung

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

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

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

\int \frac{x ^{2}}{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{2}{x + 1} - ln ∣ x ∣ + ln ∣ x + 1 ∣

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

Füge eine Konstante zur Lösung hinzu

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

x \to 0 lim (\frac{5 x ^{2}}{x ^{2}})

\int_{- 1}^{1} π (1 - x^{2})^{2} d x

d er i v a t i v e f (t) = (4 t^{2} + 5)

d er i v a t i v e f (x) = 4 t - 4 e^{t}

\int sin (21 x) d x