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Beliebt Rechnen >

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

Lösung

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

Lösung

x + 3 ln ∣ x - 1 ∣ - ln ∣ x ∣ - 1 + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

= x - 1 + ln ∣ x - 1 ∣ + ln ∣ x - 1 ∣ - ln ∣ x ∣ + ln ∣ x - 1 ∣

Vereinfache

= x + 3 ln ∣ x - 1 ∣ - ln ∣ x ∣ - 1

Füge eine Konstante zur Lösung hinzu

= x + 3 ln ∣ x - 1 ∣ - ln ∣ x ∣ - 1 + C

Graph

Beliebte Beispiele

y^{''}-10y^'+25y=t^{-7}e^{5t}

y^{^{''}} - 10 y^{^{'}} + 25 y = t^{- 7} e^{5 t}

derivative of x^2sin^7(x+xcos^{-2}(x))

\frac{d}{d x} (x^{2} sin^{7} (x) + x cos^{- 2} (x))

4y^{''}+2y^'+2y=0

4 y^{^{''}} + 2 y^{^{'}} + 2 y = 0

derivative of x+xsqrt(x)

\frac{d}{d x} (x + x x)

derivative of x^3+3^x

\frac{d}{d x} (x^{3} + 3^{x})