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Beliebt Rechnen >

integral of (x^4+x)/(x^3-x^2+x-1)

Lösung

\int \frac{x ^{4} + x}{x ^{3} - x ^{2} + x - 1} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{4} + x}{x ^{3} - x ^{2} + x - 1} d x

Lange Division

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

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

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

\int 1 d x = x

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 9xsin(2x)

\int 9 x sin (2 x) d x

integral of tan^3(x)cos^2(x)

\int tan^{3} (x) cos^{2} (x) d x

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

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

integral of (x^4)/((1+x^5)^{2/3)}

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

integral of 1/((x^2+8x+25)^{3/2)}

\int \frac{1}{( x ^{2} + 8 x + 25 ) ^{\frac{3}{2}}} d x