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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\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 )}

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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

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

integral of 3/(x(x^2+6)^2)

\int \frac{3}{x ( x ^{2} + 6 ) ^{2}} d x

integral of (3x^4+2x)/(x^3)

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integral of (sin^2(θ))/(cos^2(θ))

\int \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )} d θ

integral of (x-2)/(x^2+6x+13)

\int \frac{x - 2}{x ^{2} + 6 x + 13} d x