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Beliebt Rechnen >

integral of (5x^2+3x)/((x-1)(x+1)^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int cos^{2} (s) d s

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\int 5 - ∣ x ∣ d x

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\int \frac{t}{e ^{- t}} d t

integral of 4sin(x)+5sinh(x)

\int 4 sin (x) + 5 sinh (x) d x