de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{4} (x^{2} - 2 x + 2 ln ∣ x + 1 ∣ - 3) + C

Schritte zur Lösung

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

\frac{x ^{2}}{2 x + 2} = \frac{1}{2} \frac{x ^{2}}{x + 1}

= \int \frac{1}{2} \cdot \frac{x ^{2}}{x + 1} d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{x ^{2}}{x + 1} d x

Wende U-Substitution an

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

Multipliziere aus

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

= \frac{1}{2} \cdot \int u - 2 + \frac{1}{u} 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

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

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

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

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

Setze in

u = x + 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x^5)/(x^4-1)

\int \frac{x ^{5}}{x ^{4} - 1} d x

integral of (x^2)/(x^3-27)

\int \frac{x ^{2}}{x ^{3} - 27} d x

integral of 2xe^{15x^2}

\int 2 x e^{15 x^{2}} d x

integral of sqrt(x)(x-4)

\int x (x - 4) d x

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

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