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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Lange Division

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

= 2 \cdot \int x + 1 + \frac{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

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

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

\int 1 d x = x

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int \frac{1 - 25 x ^{2}}{x ^{2}} d x

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\int 3 \frac{cos ( x ) + sin ( x )}{sin ( 2 x )} d x

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d er i v a t i v e e^{2 x + 1}

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d er i v a t i v e f (x) = 5 x^{- 2}

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\int (3 x - 2) e - x d x