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Beliebt Rechnen >

integral of 1/(x-2x^{2/3)-3x^{1/3}}

Lösung

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

Lösung

3 (\frac{1}{4} ln x^{\frac{1}{3}} + 1 + \frac{3}{4} ln x^{\frac{1}{3}} - 3) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{3 u}{u ^{2} - 2 u - 3} d u

Entferne die Konstante:

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

= 3 \cdot \int \frac{u}{u ^{2} - 2 u - 3} d u

Ermittle den Partialbruch von

\frac{u}{u ^{2} - 2 u - 3} : \frac{1}{4 ( u + 1 )} + \frac{3}{4 ( u - 3 )}

= 3 \cdot \int \frac{1}{4 ( u + 1 )} + \frac{3}{4 ( u - 3 )} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (\int \frac{1}{4 ( u + 1 )} d u + \int \frac{3}{4 ( u - 3 )} d u)

\int \frac{1}{4 ( u + 1 )} d u = \frac{1}{4} ln ∣ u + 1 ∣

\int \frac{3}{4 ( u - 3 )} d u = \frac{3}{4} ln ∣ u - 3 ∣

= 3 (\frac{1}{4} ln ∣ u + 1 ∣ + \frac{3}{4} ln ∣ u - 3 ∣)

Setze in

u = x^{\frac{1}{3}}

ein

= 3 (\frac{1}{4} ln x^{\frac{1}{3}} + 1 + \frac{3}{4} ln x^{\frac{1}{3}} - 3)

Füge eine Konstante zur Lösung hinzu

= 3 (\frac{1}{4} ln x^{\frac{1}{3}} + 1 + \frac{3}{4} ln x^{\frac{1}{3}} - 3) + C

Graph

Beliebte Beispiele

integral of xsqrt(3x^2-5)

\int x 3 x^{2} - 5 d x

derivative of sqrt(x^2+y^2-1)

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

integral of 1/(sqrt(3x^2+6x-4))

\int \frac{1}{3 x ^{2} + 6 x - 4} d x

sum from n=1 to infinity of 1/(n+5)

n = 1 \sum \infty \frac{1}{n + 5}

(dy)/(dx)+2y=3x

\frac{d y}{d x} + 2 y = 3 x