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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

\int \frac{1}{1 - cot ( x )} d x

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

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

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

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

integral of (1/(5x))

\int (\frac{1}{5 x}) d x

integral of (x/(x^2))

\int (\frac{x}{x ^{2}}) d x