de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{4} ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣ - \frac{7}{4} ln ∣ x + 3 ∣ + \frac{3}{2 ( x + 3 )} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x}{( x + 1 ) ( x + 2 ) ( x + 3 ) ^{2}} : - \frac{1}{4 ( x + 1 )} + \frac{2}{x + 2} - \frac{7}{4 ( x + 3 )} - \frac{3}{2 ( x + 3 ) ^{2}}

= \int - \frac{1}{4 ( x + 1 )} + \frac{2}{x + 2} - \frac{7}{4 ( x + 3 )} - \frac{3}{2 ( x + 3 ) ^{2}} 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

= - \int \frac{1}{4 ( x + 1 )} d x + \int \frac{2}{x + 2} d x - \int \frac{7}{4 ( x + 3 )} d x - \int \frac{3}{2 ( x + 3 ) ^{2}} d x

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

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

\int \frac{7}{4 ( x + 3 )} d x = \frac{7}{4} ln ∣ x + 3 ∣

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

= - \frac{1}{4} ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣ - \frac{7}{4} ln ∣ x + 3 ∣ - (- \frac{3}{2 ( x + 3 )})

Vereinfache

= - \frac{1}{4} ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣ - \frac{7}{4} ln ∣ x + 3 ∣ + \frac{3}{2 ( x + 3 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} ln ∣ x + 1 ∣ + 2 ln ∣ x + 2 ∣ - \frac{7}{4} ln ∣ x + 3 ∣ + \frac{3}{2 ( x + 3 )} + C

Graph

Beliebte Beispiele

y^{^{'}} + 2 y = e^{t}

integral of (3-x)7^{(3-x)^2}

\int (3 - x) 7^{(3 - x)^{2}} d x

y^'=(7x^4e^{y/x}+x^2y^2)/(x^3y)

y^{^{'}} = \frac{7 x ^{4} e ^{\frac{y}{x}} + x ^{2} y ^{2}}{x ^{3} y}

derivative of 3/2 tan(x)

\frac{d}{d x} (\frac{3}{2} tan (x))

integral of y/(y^2+9)

\int \frac{y}{y ^{2} + 9} d y