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Beliebt Rechnen >

integral of (48x^2)/((x-18)(x+6)^2)

Lösung

\int \frac{48 x ^{2}}{( x - 18 ) ( x + 6 ) ^{2}} d x

Lösung

48 (\frac{9}{16} ln ∣ x - 18 ∣ + \frac{7}{16} ln ∣ x + 6 ∣ + \frac{3}{2 ( x + 6 )}) + C

Schritte zur Lösung

\int \frac{48 x ^{2}}{( x - 18 ) ( x + 6 ) ^{2}} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 48 \cdot \int \frac{x ^{2}}{( x - 18 ) ( x + 6 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{2}}{( x - 18 ) ( x + 6 ) ^{2}} : \frac{9}{16 ( x - 18 )} + \frac{7}{16 ( x + 6 )} - \frac{3}{2 ( x + 6 ) ^{2}}

= 48 \cdot \int \frac{9}{16 ( x - 18 )} + \frac{7}{16 ( x + 6 )} - \frac{3}{2 ( x + 6 ) ^{2}} 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

= 48 (\int \frac{9}{16 ( x - 18 )} d x + \int \frac{7}{16 ( x + 6 )} d x - \int \frac{3}{2 ( x + 6 ) ^{2}} d x)

\int \frac{9}{16 ( x - 18 )} d x = \frac{9}{16} ln ∣ x - 18 ∣

\int \frac{7}{16 ( x + 6 )} d x = \frac{7}{16} ln ∣ x + 6 ∣

\int \frac{3}{2 ( x + 6 ) ^{2}} d x = - \frac{3}{2 ( x + 6 )}

= 48 (\frac{9}{16} ln ∣ x - 18 ∣ + \frac{7}{16} ln ∣ x + 6 ∣ - (- \frac{3}{2 ( x + 6 )}))

Vereinfache

= 48 (\frac{9}{16} ln ∣ x - 18 ∣ + \frac{7}{16} ln ∣ x + 6 ∣ + \frac{3}{2 ( x + 6 )})

Füge eine Konstante zur Lösung hinzu

= 48 (\frac{9}{16} ln ∣ x - 18 ∣ + \frac{7}{16} ln ∣ x + 6 ∣ + \frac{3}{2 ( x + 6 )}) + C

Graph

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