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

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

in v erse l a pl a ce \frac{s e ^{- 2 s}}{s ^{2} + 3 s + 2}

x \to \frac{1}{2} lim (\frac{6 x - 1}{x})

\int x^{14} e^{x^{3}} d x

\frac{\partial}{\partial x} (e^{x l n (y)})

x \to 2 lim (\frac{x}{( x - 2 ) ^{2}})