de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x + 1 ∣ - 2 ln ∣ x + 2 ∣ + ln ∣ x + 3 ∣ + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

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

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

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

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

Vereinfache

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

= ln ∣ x + 1 ∣ - 2 ln ∣ x + 2 ∣ + ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 1 ∣ - 2 ln ∣ x + 2 ∣ + ln ∣ x + 3 ∣ + C

Graph

Beliebte Beispiele

integral of x(3x^2+5)^9

\int x (3 x^{2} + 5)^{9} d x

integral of 2/(vsqrt(v^2-16))

\int \frac{2}{v v ^{2} - 16} d v

integral 1/(u^2)

in t e g r a l \frac{1}{u ^{2}}

integral of 1/(sin^5(x)cos^5(x))

\int \frac{1}{sin ^{5} ( x ) cos ^{5} ( x )} d x

integral of (e^{3x})/(e^{3x)+5}

\int \frac{e ^{3 x}}{e ^{3 x} + 5} d x