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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

= 3 (\frac{1}{2} ln ∣ x - 1 ∣ + \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))

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of e^xsin(y)

\int e^{x} sin (y) d x

derivative of y^2-2y*cos(x)

\frac{d}{d x} (y^{2} - 2 y \cdot cos (x))

(1+cos(t))y^'=(1+e-y)sin(t),y(0)=0

(1 + cos (t)) y^{^{'}} = (1 + e - y) sin (t), y (0) = 0

derivative (x-2)^2(x-1)

d er i v a t i v e (x - 2)^{2} (x - 1)

(dv)/(dt)=0.64-v/(29400)*8,v(0)=0

\frac{d v}{d t} = 0.64 - \frac{v}{29400} \cdot 8, v (0) = 0