integral of (x^3)/((x^2+16)^3)

Lösung

\int \frac{x ^{3}}{( x ^{2} + 16 ) ^{3}} d x

- \frac{1}{2 ( x ^{2} + 16 )} + \frac{4}{( x ^{2} + 16 ) ^{2}} + C

Schritte zur Lösung

\int \frac{x ^{3}}{( x ^{2} + 16 ) ^{3}} d x

Ermittle den Partialbruch von

\frac{x ^{3}}{( x ^{2} + 16 ) ^{3}} : \frac{x}{( x ^{2} + 16 ) ^{2}} - \frac{16 x}{( x ^{2} + 16 ) ^{3}}

= \int \frac{x}{( x ^{2} + 16 ) ^{2}} - \frac{16 x}{( x ^{2} + 16 ) ^{3}} 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{x}{( x ^{2} + 16 ) ^{2}} d x - \int \frac{16 x}{( x ^{2} + 16 ) ^{3}} d x

\int \frac{x}{( x ^{2} + 16 ) ^{2}} d x = - \frac{1}{2 ( x ^{2} + 16 )}

\int \frac{16 x}{( x ^{2} + 16 ) ^{3}} d x = - \frac{4}{( x ^{2} + 16 ) ^{2}}

= - \frac{1}{2 ( x ^{2} + 16 )} - (- \frac{4}{( x ^{2} + 16 ) ^{2}})

Vereinfache

= - \frac{1}{2 ( x ^{2} + 16 )} + \frac{4}{( x ^{2} + 16 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2 ( x ^{2} + 16 )} + \frac{4}{( x ^{2} + 16 ) ^{2}} + C

(- x y^{2} - y) d x + (- x^{2} y - x) d y = 0

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