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

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int \frac{x ^{2} ( - x + 2 )}{( x + 1 ) ^{4}} d x = - ln ∣ x + 1 ∣ - \frac{3}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 1 ) ^{3}} - \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} + \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

l a pl a ce t r an s f or m 5 y

\frac{d}{d x} (\frac{d y}{d x}) + y = 7 sin (x)

\int \frac{0.8 ( 3 + ln ( u ) ) ^{3}}{u} d u

t an g e n t y = 2 x^{3} - 8 x, (2, 0)

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