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Beliebt Rechnen >

integral of (2x^2-5x+4)/((x-2)^3)

Lösung

\int \frac{2 x ^{2} - 5 x + 4}{( x - 2 ) ^{3}} d x

Lösung

- \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ + \frac{1}{x - 2} + \frac{3}{( x - 2 ) ^{2}} + C

Schritte zur Lösung

\int \frac{2 x ^{2} - 5 x + 4}{( x - 2 ) ^{3}} d x

Multipliziere aus

\frac{2 x ^{2} - 5 x + 4}{( x - 2 ) ^{3}} : \frac{2 x ^{2}}{( x - 2 ) ^{3}} - \frac{5 x}{( x - 2 ) ^{3}} + \frac{4}{( x - 2 ) ^{3}}

= \int \frac{2 x ^{2}}{( x - 2 ) ^{3}} - \frac{5 x}{( x - 2 ) ^{3}} + \frac{4}{( x - 2 ) ^{3}} 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{2 x ^{2}}{( x - 2 ) ^{3}} d x - \int \frac{5 x}{( x - 2 ) ^{3}} d x + \int \frac{4}{( x - 2 ) ^{3}} d x

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

\int \frac{5 x}{( x - 2 ) ^{3}} d x = 5 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}})

\int \frac{4}{( x - 2 ) ^{3}} d x = - \frac{2}{( x - 2 ) ^{2}}

= - \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ - \frac{4}{x - 2} - 5 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) - \frac{2}{( x - 2 ) ^{2}}

Vereinfache

- \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ - \frac{4}{x - 2} - 5 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) - \frac{2}{( x - 2 ) ^{2}} : - \frac{x ^{2}}{( x - 2 ) ^{2}} + 2 ln ∣ x - 2 ∣ + \frac{1}{x - 2} + \frac{3}{( x - 2 ) ^{2}}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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

\int 2 x (x^{2} - 3)^{3} d x

integral of x^k

\int x^{k} d x

derivative (sqrt(x)+2)(x^3-x)

d er i v a t i v e (x + 2) (x^{3} - x)

derivative of (x^3/(sqrt(x)))

\frac{d}{d x} (\frac{x ^{3}}{x})

integral of 1/((sqrt(16-x^2))^3)

\int \frac{1}{( 16 - x ^{2} ) ^{3}} d x