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Beliebt Rechnen >

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

Lösung

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

Lösung

3 ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{5}{x - 1} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

= 3 ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{5}{x - 1}

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x ∣ - ln ∣ x - 1 ∣ - \frac{5}{x - 1} + C

Graph

Beliebte Beispiele

integral of ((3x+4))/(x+3)

\int \frac{( 3 x + 4 )}{x + 3} d x

(\partial}{\partial y}(\frac{(x-y))/x)

\frac{\partial}{\partial y} (\frac{( x - y )}{x})

y^{'} = y^{3} - y

derivative of 3x^4+1/(3x^4+3/(2x^2))

\frac{d}{d x} (3 x^{4} + \frac{1}{3 x ^{4}} + \frac{3}{2 x ^{2}})

integral from-5 to-2 of 4x^3

\int_{- 5}^{- 2} 4 x^{3} d x