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Beliebt Rechnen >

integral of 1/((x+2)(x+7)(2x-5)(x+1)^2)

Lösung

\int \frac{1}{( x + 2 ) ( x + 7 ) ( 2 x - 5 ) ( x + 1 ) ^{2}} d x

Lösung

- \frac{1}{45} ln ∣ x + 2 ∣ + \frac{1}{3420} ln ∣ x + 7 ∣ + \frac{8}{8379} ln ∣ 2 x - 5 ∣ + \frac{37}{1764} ln ∣ x + 1 ∣ + \frac{1}{42 ( x + 1 )} + C

Schritte zur Lösung

\int \frac{1}{( x + 2 ) ( x + 7 ) ( 2 x - 5 ) ( x + 1 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{1}{( x + 2 ) ( x + 7 ) ( 2 x - 5 ) ( x + 1 ) ^{2}} : - \frac{1}{45 ( x + 2 )} + \frac{1}{3420 ( x + 7 )} + \frac{16}{8379 ( 2 x - 5 )} + \frac{37}{1764 ( x + 1 )} - \frac{1}{42 ( x + 1 ) ^{2}}

= \int - \frac{1}{45 ( x + 2 )} + \frac{1}{3420 ( x + 7 )} + \frac{16}{8379 ( 2 x - 5 )} + \frac{37}{1764 ( x + 1 )} - \frac{1}{42 ( x + 1 ) ^{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{1}{45 ( x + 2 )} d x + \int \frac{1}{3420 ( x + 7 )} d x + \int \frac{16}{8379 ( 2 x - 5 )} d x + \int \frac{37}{1764 ( x + 1 )} d x - \int \frac{1}{42 ( x + 1 ) ^{2}} d x

\int \frac{1}{45 ( x + 2 )} d x = \frac{1}{45} ln ∣ x + 2 ∣

\int \frac{1}{3420 ( x + 7 )} d x = \frac{1}{3420} ln ∣ x + 7 ∣

\int \frac{16}{8379 ( 2 x - 5 )} d x = \frac{8}{8379} ln ∣ 2 x - 5 ∣

\int \frac{37}{1764 ( x + 1 )} d x = \frac{37}{1764} ln ∣ x + 1 ∣

\int \frac{1}{42 ( x + 1 ) ^{2}} d x = - \frac{1}{42 ( x + 1 )}

= - \frac{1}{45} ln ∣ x + 2 ∣ + \frac{1}{3420} ln ∣ x + 7 ∣ + \frac{8}{8379} ln ∣ 2 x - 5 ∣ + \frac{37}{1764} ln ∣ x + 1 ∣ - (- \frac{1}{42 ( x + 1 )})

Vereinfache

= - \frac{1}{45} ln ∣ x + 2 ∣ + \frac{1}{3420} ln ∣ x + 7 ∣ + \frac{8}{8379} ln ∣ 2 x - 5 ∣ + \frac{37}{1764} ln ∣ x + 1 ∣ + \frac{1}{42 ( x + 1 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{45} ln ∣ x + 2 ∣ + \frac{1}{3420} ln ∣ x + 7 ∣ + \frac{8}{8379} ln ∣ 2 x - 5 ∣ + \frac{37}{1764} ln ∣ x + 1 ∣ + \frac{1}{42 ( x + 1 )} + C

Graph

Beliebte Beispiele

fläche y=xe^{-0.4x},x=5,y=0

a re a y = x e^{- 0.4 x}, x = 5, y = 0

derivative of 2+sin(x)

\frac{d}{d x} (2 + sin (x))

limit as x approaches 5+of (x-5)/(x-5)

x \to 5 + lim (\frac{x - 5}{x - 5})

limit as x approaches 0 of sqrt(x)ln(x)

x \to 0 lim (x ln (x))

xy^'-y=2x^2-x-2

x y^{^{'}} - y = 2 x^{2} - x - 2