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Beliebt Rechnen >

integral of 1/(9x^3-2x^2)

Lösung

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

Lösung

- \frac{9}{4} ln ∣ x ∣ + \frac{1}{2 x} + \frac{9}{4} ln ∣ 9 x - 2 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{1}{9 x ^{3} - 2 x ^{2}} : - \frac{9}{4 x} - \frac{1}{2 x ^{2}} + \frac{81}{4 ( 9 x - 2 )}

= \int - \frac{9}{4 x} - \frac{1}{2 x ^{2}} + \frac{81}{4 ( 9 x - 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

= - \int \frac{9}{4 x} d x - \int \frac{1}{2 x ^{2}} d x + \int \frac{81}{4 ( 9 x - 2 )} d x

\int \frac{9}{4 x} d x = \frac{9}{4} ln ∣ x ∣

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

\int \frac{81}{4 ( 9 x - 2 )} d x = \frac{9}{4} ln ∣ 9 x - 2 ∣

= - \frac{9}{4} ln ∣ x ∣ - (- \frac{1}{2 x}) + \frac{9}{4} ln ∣ 9 x - 2 ∣

Vereinfache

= - \frac{9}{4} ln ∣ x ∣ + \frac{1}{2 x} + \frac{9}{4} ln ∣ 9 x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{9}{4} ln ∣ x ∣ + \frac{1}{2 x} + \frac{9}{4} ln ∣ 9 x - 2 ∣ + C

Graph

Beliebte Beispiele

tangent y=x^2-3x+3,(0,3)

t an g e n t y = x^{2} - 3 x + 3, (0, 3)

derivative f(x)= A/(x^7)+Be^7

d er i v a t i v e f (x) = \frac{A}{x ^{7}} + B e^{7}

derivative of cos(2)

\frac{d}{d x} (cos (2))

integral from 1 to 3 of (x^2+1)^3

\int_{1}^{3} (x^{2} + 1)^{3} d x

(d^2)/(dx^2)(8xcos(x^2))

\frac{d ^{2}}{d x ^{2}} (8 x cos (x^{2}))