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Beliebt Rechnen >

integral of (81)/(x^3-9x^2)

Lösung

\int \frac{81}{x ^{3} - 9 x ^{2}} d x

Lösung

- ln ∣ x ∣ + \frac{9}{x} + ln ∣ x - 9 ∣ + C

Schritte zur Lösung

\int \frac{81}{x ^{3} - 9 x ^{2}} d x

Entferne die Konstante:

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

= 81 \cdot \int \frac{1}{x ^{3} - 9 x ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{x ^{3} - 9 x ^{2}} : - \frac{1}{81 x} - \frac{1}{9 x ^{2}} + \frac{1}{81 ( x - 9 )}

= 81 \cdot \int - \frac{1}{81 x} - \frac{1}{9 x ^{2}} + \frac{1}{81 ( x - 9 )} 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

= 81 (- \int \frac{1}{81 x} d x - \int \frac{1}{9 x ^{2}} d x + \int \frac{1}{81 ( x - 9 )} d x)

\int \frac{1}{81 x} d x = \frac{1}{81} ln ∣ x ∣

\int \frac{1}{9 x ^{2}} d x = - \frac{1}{9 x}

\int \frac{1}{81 ( x - 9 )} d x = \frac{1}{81} ln ∣ x - 9 ∣

= 81 (- \frac{1}{81} ln ∣ x ∣ - (- \frac{1}{9 x}) + \frac{1}{81} ln ∣ x - 9 ∣)

Vereinfache

81 (- \frac{1}{81} ln ∣ x ∣ - (- \frac{1}{9 x}) + \frac{1}{81} ln ∣ x - 9 ∣) : - ln ∣ x ∣ + \frac{9}{x} + ln ∣ x - 9 ∣

= - ln ∣ x ∣ + \frac{9}{x} + ln ∣ x - 9 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ x ∣ + \frac{9}{x} + ln ∣ x - 9 ∣ + C

Graph

Beliebte Beispiele

tangent (x^2+1)/(x^2+x+1)

t an g e n t \frac{x ^{2} + 1}{x ^{2} + x + 1}

derivative f(x)=sqrt(7+10x)

d er i v a t i v e f (x) = 7 + 10 x

fläche y=sqrt(x)+1,y=(x^2)/4 ,x=0,x=1

a re a y = x + 1, y = \frac{x ^{2}}{4}, x = 0, x = 1

y^'=(x^2-3y^2)/(2xy)

y^{^{'}} = \frac{x ^{2} - 3 y ^{2}}{2 x y}

limit as x approaches 11 of sqrt(3)

x \to 11 lim (3)