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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

= \int - \frac{1}{u} + \frac{1}{u ^{2}} + \frac{1}{u + 1} d 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{1}{u} d u + \int \frac{1}{u ^{2}} d u + \int \frac{1}{u + 1} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= - ln ∣ u ∣ - \frac{1}{u} + ln ∣ u + 1 ∣

Setze in

u = x + 1

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 1/(8x-3)

\int \frac{1}{8 x - 3} d x

integral of 6x*sqrt(x^2+4)

\int 6 x \cdot x^{2} + 4 d x

integral of ((36-x^2))/(144)

\int \frac{( 36 - x ^{2} )}{144} d x

integral of 2/(x^7)

\int \frac{2}{x ^{7}} d x

integral of (sqrt(2)-sqrt(x))^2

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