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Beliebt Rechnen >

integral of (4x)/((x-1)^2(x+1))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative of (x^2+3^5)

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

derivative f(x)=(sin(x))^2

d er i v a t i v e f (x) = (sin (x))^{2}

fläche x^2+1,0,2

a re a x^{2} + 1, 0, 2

(\partial)/(\partial x)(f(x,y)(xy))

\frac{\partial}{\partial x} (f (x, y) (x y))

limit as x approaches 0-of (4-x)/x

x \to 0 - lim (\frac{4 - x}{x})