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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x+1)/(x^2-6x+8)

\int \frac{x + 1}{x ^{2} - 6 x + 8} d x

integral of (x^4)/(x^5-7)

\int \frac{x ^{4}}{x ^{5} - 7} d x

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\int x sin (x^{2}) cos (x^{2}) d x

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\int e^{t} + 1 d t

integral of (1+cos^2(θ))/(cos^2(θ))

\int \frac{1 + cos ^{2} ( θ )}{cos ^{2} ( θ )} d θ