de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of (3x+5)/((x+1)(x-1)^2)

Lösung

\int \frac{3 x + 5}{( x + 1 ) ( x - 1 ) ^{2}} d x

Lösung

\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ - \frac{4}{x - 1} + C

Schritte zur Lösung

\int \frac{3 x + 5}{( x + 1 ) ( x - 1 ) ^{2}} d x

Ermittle den Partialbruch von

\frac{3 x + 5}{( x + 1 ) ( x - 1 ) ^{2}} : \frac{1}{2 ( x + 1 )} - \frac{1}{2 ( x - 1 )} + \frac{4}{( x - 1 ) ^{2}}

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

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

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

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

= \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ - \frac{4}{x - 1}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ - \frac{4}{x - 1} + C

Graph

Beliebte Beispiele

integral of 1/(sqrt(20+x^2))

\int \frac{1}{20 + x ^{2}} d x

integral of y^4ln(sqrt(y))

\int y^{4} ln (y) d y

integral of 1/(xln(x)ln^2(ln(x)))

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

integral of (0.1q+1)e^{0.3q}

\int (0.1 q + 1) e^{0.3 q} d q

integral of (2x^2-13x+11)/(x-1)

\int \frac{2 x ^{2} - 13 x + 11}{x - 1} d x