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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{4} ln ∣ x - 1 ∣ + \frac{1}{4} (\frac{1}{2} ln \frac{1}{4} (x - 1)^{2} + 1 + 2 arctan (\frac{1}{2} (x - 1))) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \int - \frac{1}{4 ( x - 1 )} + \frac{x + 3}{4 ( x ^{2} - 2 x + 5 )} 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}{4 ( x - 1 )} d x + \int \frac{x + 3}{4 ( x ^{2} - 2 x + 5 )} d x

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

\int \frac{x + 3}{4 ( x ^{2} - 2 x + 5 )} d x = \frac{1}{4} (\frac{1}{2} ln \frac{1}{4} (x - 1)^{2} + 1 + 2 arctan (\frac{1}{2} (x - 1)))

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of x/((x+2)^{1/4)}

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

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

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

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

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

integral of (9x^4)/(4x^5+3)

\int \frac{9 x ^{4}}{4 x ^{5} + 3} d x

integral of 23e^u

\int 23 e^{u} d u