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Beliebt Rechnen >

integral of (x^4-x+1)/(x^4(2x-1))

Lösung

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

Lösung

- 4 ln ∣ x ∣ + \frac{2}{x} + \frac{1}{2 x ^{2}} + \frac{1}{3 x ^{3}} + \frac{9}{2} ln ∣ 2 x - 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \int - \frac{4}{x} - \frac{2}{x ^{2}} - \frac{1}{x ^{3}} - \frac{1}{x ^{4}} + \frac{9}{2 x - 1} 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{4}{x} d x - \int \frac{2}{x ^{2}} d x - \int \frac{1}{x ^{3}} d x - \int \frac{1}{x ^{4}} d x + \int \frac{9}{2 x - 1} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

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

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

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

= - 4 ln ∣ x ∣ - (- \frac{2}{x}) - (- \frac{1}{2 x ^{2}}) - (- \frac{1}{3 x ^{3}}) + \frac{9}{2} ln ∣ 2 x - 1 ∣

Vereinfache

= - 4 ln ∣ x ∣ + \frac{2}{x} + \frac{1}{2 x ^{2}} + \frac{1}{3 x ^{3}} + \frac{9}{2} ln ∣ 2 x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= - 4 ln ∣ x ∣ + \frac{2}{x} + \frac{1}{2 x ^{2}} + \frac{1}{3 x ^{3}} + \frac{9}{2} ln ∣ 2 x - 1 ∣ + C

Graph

Beliebte Beispiele

integral of 1/(shx)

\int \frac{1}{s h x} d x

integral of (x^2-2x+1)/(4(x-1))

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

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

\int \frac{x ^{3} - 2 x}{x ^{3} + x} d x

integral of (e^{arctan(x)}-2x)/(1+x^2)

\int \frac{e ^{a r c t a n (x)} - 2 x}{1 + x ^{2}} d x

integral of (x+4)e^{-2x}

\int (x + 4) e^{- 2 x} d x