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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x - 1 ∣ - \frac{3}{x - 1} + C

Schritte zur Lösung

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

Multipliziere aus

\frac{x + 2}{x ^{2} - 2 x + 1} : \frac{x}{x ^{2} - 2 x + 1} + \frac{2}{x ^{2} - 2 x + 1}

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

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

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

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

Vereinfache

ln ∣ x - 1 ∣ - \frac{1}{x - 1} - \frac{2}{x - 1} : ln ∣ x - 1 ∣ - \frac{3}{x - 1}

= ln ∣ x - 1 ∣ - \frac{3}{x - 1}

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 1 ∣ - \frac{3}{x - 1} + C

Graph

Beliebte Beispiele

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

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

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

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

integral of 5/(1+e^x)

\int \frac{5}{1 + e ^{x}} d x

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

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

integral of ((-2x+6))/((x^2+2x+4)^2)

\int \frac{( - 2 x + 6 )}{( x ^{2} + 2 x + 4 ) ^{2}} d x