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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

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

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

Setze in

u = x + 1

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

\frac{d y}{d x} = 2 y + 1

sum from n=0 to infinity of 2/(1.15^n)

n = 0 \sum \infty \frac{2}{1.1 5 ^{n}}

y^{''}+25y=-30sec(5t)

y^{^{''}} + 25 y = - 30 sec (5 t)

derivative of (3x^2-6/(2x+3))

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

derivative of e^x*cos(2x)

\frac{d}{d x} (e^{x} \cdot cos (2 x))