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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int \frac{x ^{2} + 2 x + 3}{x ^{2} - 6 x + 8} d x

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\int 1 + u d u

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\int \frac{lo g _{e} ( x )}{x ^{2}} d x

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

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