de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral x/(((x-1)(x-2)(x-3)))

Lösung

integral

\frac{x}{( ( x - 1 ) ( x - 2 ) ( x - 3 ) )}

Lösung

\frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ - 2 ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{u + 2}{u ( u + 1 ) ( u - 1 )} d u

Multipliziere aus

\frac{u + 2}{u ( u + 1 ) ( u - 1 )} : \frac{u}{u ( u + 1 ) ( u - 1 )} + \frac{2}{u ( u + 1 ) ( u - 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{u}{u ( u + 1 ) ( u - 1 )} d u + \int \frac{2}{u ( u + 1 ) ( u - 1 )} d u

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

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

= - \frac{1}{2} ln ∣ u + 1 ∣ + \frac{1}{2} ln ∣ u - 1 ∣ - 2 ln ∣ u ∣ + ln ∣ u + 1 ∣ + ln ∣ u - 1 ∣

Setze in

u = x - 2

ein

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

Vereinfache

- \frac{1}{2} ln ∣ x - 2 + 1 ∣ + \frac{1}{2} ln ∣ x - 2 - 1 ∣ - 2 ln ∣ x - 2 ∣ + ln ∣ x - 2 + 1 ∣ + ln ∣ x - 2 - 1 ∣ : \frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ - 2 ln ∣ x - 2 ∣

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln ∣ x - 3 ∣ - 2 ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral (a-x^2)^{1/2}

in t e g r a l (a - x^{2})^{\frac{1}{2}}

integral sin^3(x)cos^3(x)

in t e g r a l sin^{3} (x) cos^{3} (x)

f(x)=(((x-1))/((x+1)))^{1/2}

f (x) = (\frac{( x - 1 )}{( x + 1 )})^{\frac{1}{2}}

integral sin^2(x)cos^3(x)

in t e g r a l sin^{2} (x) cos^{3} (x)

integral 1/(v^4)

in t e g r a l \frac{1}{v ^{4}}