integral of (x^2)/((x+2)^6)

Lösung

\int \frac{x ^{2}}{( x + 2 ) ^{6}} d x

- \frac{1}{3 ( x + 2 ) ^{3}} - \frac{4 x}{( x + 2 ) ^{5}} + \frac{5}{( x + 2 ) ^{4}} - \frac{8}{( x + 2 ) ^{5}} - \frac{4}{5 ( x + 2 ) ^{5}} + C

Schritte zur Lösung

\int \frac{x ^{2}}{( x + 2 ) ^{6}} d x

Ermittle den Partialbruch von

\frac{x ^{2}}{( x + 2 ) ^{6}} : \frac{1}{( x + 2 ) ^{4}} - \frac{4}{( x + 2 ) ^{5}} + \frac{4}{( x + 2 ) ^{6}}

= \int \frac{1}{( x + 2 ) ^{4}} - \frac{4}{( x + 2 ) ^{5}} + \frac{4}{( x + 2 ) ^{6}} 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{1}{( x + 2 ) ^{4}} d x - \int \frac{4}{( x + 2 ) ^{5}} d x + \int \frac{4}{( x + 2 ) ^{6}} d x

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

\int \frac{4}{( x + 2 ) ^{5}} d x = \frac{4 x}{( x + 2 ) ^{5}} - \frac{5}{( x + 2 ) ^{4}} + \frac{8}{( x + 2 ) ^{5}}

\int \frac{4}{( x + 2 ) ^{6}} d x = - \frac{4}{5 ( x + 2 ) ^{5}}

= - \frac{1}{3 ( x + 2 ) ^{3}} - (\frac{4 x}{( x + 2 ) ^{5}} - \frac{5}{( x + 2 ) ^{4}} + \frac{8}{( x + 2 ) ^{5}}) - \frac{4}{5 ( x + 2 ) ^{5}}

Vereinfache

= - \frac{1}{3 ( x + 2 ) ^{3}} - \frac{4 x}{( x + 2 ) ^{5}} + \frac{5}{( x + 2 ) ^{4}} - \frac{8}{( x + 2 ) ^{5}} - \frac{4}{5 ( x + 2 ) ^{5}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3 ( x + 2 ) ^{3}} - \frac{4 x}{( x + 2 ) ^{5}} + \frac{5}{( x + 2 ) ^{4}} - \frac{8}{( x + 2 ) ^{5}} - \frac{4}{5 ( x + 2 ) ^{5}} + C

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

t \to 0 lim (\frac{1 - cos ( 3 t )}{t})

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

\int cos (t^{3}) d t

\frac{d}{d x} (ln (9) x^{6})