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Beliebt Rechnen >

integral of (48x^2)/((x-15)(x+5)^2)

Lösung

\int \frac{48 x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} d x

Lösung

48 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ + \frac{5}{4 ( x + 5 )}) + C

Schritte zur Lösung

\int \frac{48 x ^{2}}{( x - 15 ) ( x + 5 ) ^{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 - 15 ) ( x + 5 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} : \frac{9}{16 ( x - 15 )} + \frac{7}{16 ( x + 5 )} - \frac{5}{4 ( x + 5 ) ^{2}}

= 48 \cdot \int \frac{9}{16 ( x - 15 )} + \frac{7}{16 ( x + 5 )} - \frac{5}{4 ( x + 5 ) ^{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 - 15 )} d x + \int \frac{7}{16 ( x + 5 )} d x - \int \frac{5}{4 ( x + 5 ) ^{2}} d x)

\int \frac{9}{16 ( x - 15 )} d x = \frac{9}{16} ln ∣ x - 15 ∣

\int \frac{7}{16 ( x + 5 )} d x = \frac{7}{16} ln ∣ x + 5 ∣

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

= 48 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ - (- \frac{5}{4 ( x + 5 )}))

Vereinfache

= 48 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ + \frac{5}{4 ( x + 5 )})

Füge eine Konstante zur Lösung hinzu

= 48 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ + \frac{5}{4 ( x + 5 )}) + C

Graph

Beliebte Beispiele

\frac{d}{d t} (- 6 t)

inverselaplace (2x)/(x^2+2x)+x+5

in v erse l a pl a ce \frac{2 x}{x ^{2} + 2 x} + x + 5

y^{''}-14y^'+48y=0

y^{^{''}} - 14 y^{^{'}} + 48 y = 0

integral of sqrt(x+10)

\int x + 10 d x

integral from 0 to pi/2 of cos(5t)cos(10t)

\int_{0}^{\frac{π}{2}} cos (5 t) cos (10 t) d t