integral of (x^3)/((x+3)^4)

Lösung

\int \frac{x ^{3}}{( x + 3 ) ^{4}} d x

ln ∣ x + 3 ∣ + \frac{9}{x + 3} - \frac{27}{2 ( x + 3 ) ^{2}} + \frac{9}{( x + 3 ) ^{3}} + C

Schritte zur Lösung

\int \frac{x ^{3}}{( x + 3 ) ^{4}} d x

Ermittle den Partialbruch von

\frac{x ^{3}}{( x + 3 ) ^{4}} : \frac{1}{x + 3} - \frac{9}{( x + 3 ) ^{2}} + \frac{27}{( x + 3 ) ^{3}} - \frac{27}{( x + 3 ) ^{4}}

= \int \frac{1}{x + 3} - \frac{9}{( x + 3 ) ^{2}} + \frac{27}{( x + 3 ) ^{3}} - \frac{27}{( x + 3 ) ^{4}} 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 + 3} d x - \int \frac{9}{( x + 3 ) ^{2}} d x + \int \frac{27}{( x + 3 ) ^{3}} d x - \int \frac{27}{( x + 3 ) ^{4}} d x

\int \frac{1}{x + 3} d x = ln ∣ x + 3 ∣

\int \frac{9}{( x + 3 ) ^{2}} d x = - \frac{9}{x + 3}

\int \frac{27}{( x + 3 ) ^{3}} d x = - \frac{27}{2 ( x + 3 ) ^{2}}

\int \frac{27}{( x + 3 ) ^{4}} d x = - \frac{9}{( x + 3 ) ^{3}}

= ln ∣ x + 3 ∣ - (- \frac{9}{x + 3}) - \frac{27}{2 ( x + 3 ) ^{2}} - (- \frac{9}{( x + 3 ) ^{3}})

Vereinfache

= ln ∣ x + 3 ∣ + \frac{9}{x + 3} - \frac{27}{2 ( x + 3 ) ^{2}} + \frac{9}{( x + 3 ) ^{3}}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 3 ∣ + \frac{9}{x + 3} - \frac{27}{2 ( x + 3 ) ^{2}} + \frac{9}{( x + 3 ) ^{3}} + C

\int A^{2} d A

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

d er i v a t i v e 5 t + 6 t^{5}

\frac{d}{d x} (x (1 - x))

y^{^{'}} + 2 y = 0