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^2+5x-4)/((x^2-1)(x+1))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{3 u ^{2} - u - 6}{u ( u ^{2} - 2 u )} d u

Multipliziere aus

\frac{3 u ^{2} - u - 6}{u ( u ^{2} - 2 u )} : \frac{3 u ^{2}}{u ( u ^{2} - 2 u )} - \frac{u}{u ( u ^{2} - 2 u )} - \frac{6}{u ( u ^{2} - 2 u )}

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{3 u ^{2}}{u ( u ^{2} - 2 u )} d u - \int \frac{u}{u ( u ^{2} - 2 u )} d u - \int \frac{6}{u ( u ^{2} - 2 u )} d u

\int \frac{3 u ^{2}}{u ( u ^{2} - 2 u )} d u = 3 ln ∣ u - 2 ∣

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

\int \frac{6}{u ( u ^{2} - 2 u )} d u = 6 (- \frac{1}{4} ln ∣ u ∣ + \frac{1}{2 u} + \frac{1}{4} ln ∣ u - 2 ∣)

= 3 ln ∣ u - 2 ∣ - (- \frac{1}{2} ln ∣ u ∣ + \frac{1}{2} ln ∣ u - 2 ∣) - 6 (- \frac{1}{4} ln ∣ u ∣ + \frac{1}{2 u} + \frac{1}{4} ln ∣ u - 2 ∣)

Setze in

u = x + 1

ein

= 3 ln ∣ x + 1 - 2 ∣ - (- \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x + 1 - 2 ∣) - 6 (- \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{2 ( x + 1 )} + \frac{1}{4} ln ∣ x + 1 - 2 ∣)

Vereinfache

3 ln ∣ x + 1 - 2 ∣ - (- \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2} ln ∣ x + 1 - 2 ∣) - 6 (- \frac{1}{4} ln ∣ x + 1 ∣ + \frac{1}{2 ( x + 1 )} + \frac{1}{4} ln ∣ x + 1 - 2 ∣) : 2 ln ∣ x + 1 ∣ + 3 ln ∣ x - 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ - \frac{3}{x + 1} - \frac{3}{2} ln ∣ x - 1 ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 2/(3\sqrt[3]{x)}

\int \frac{2}{3 3 x} d x

(\partial)/(\partial x)(-4sin(x))

\frac{\partial}{\partial x} (- 4 sin (x))

derivative of ((-3x+1^4-5)^3)

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

derivative 2-2x^2

d er i v a t i v e 2 - 2 x^{2}

integral of y/((y^2+1)^2)

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