de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

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

Lösung

integral

\frac{x ^{3}}{x - 2}

Lösung

3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{( u + 2 ) ^{3}}{u} d u

Multipliziere aus

\frac{( u + 2 ) ^{3}}{u} : u^{2} + 6 u + 12 + \frac{8}{u}

= \int u^{2} + 6 u + 12 + \frac{8}{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

= \int u^{2} d u + \int 6 u d u + \int 12 d u + \int \frac{8}{u} d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int 6 u d u = 3 u^{2}

\int 12 d u = 12 u

\int \frac{8}{u} d u = 8 ln ∣ u ∣

= \frac{u ^{3}}{3} + 3 u^{2} + 12 u + 8 ln ∣ u ∣

Setze in

u = x - 2

ein

= \frac{( x - 2 ) ^{3}}{3} + 3 (x - 2)^{2} + 12 (x - 2) + 8 ln ∣ x - 2 ∣

Vereinfache

\frac{( x - 2 ) ^{3}}{3} + 3 (x - 2)^{2} + 12 (x - 2) + 8 ln ∣ x - 2 ∣ : 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12

= 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12

Füge eine Konstante zur Lösung hinzu

= 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 + C

Graph

Beliebte Beispiele

integral (e^3)/x

in t e g r a l \frac{e ^{3}}{x}

integral (x^{0.5})/((1+x)^2)

in t e g r a l \frac{x ^{0.5}}{( 1 + x ) ^{2}}

integral x/(((x-1)(x-2)(x-3)))

in t e g r a l \frac{x}{( ( x - 1 ) ( x - 2 ) ( x - 3 ) )}

integral (a-x^2)^{1/2}

in t e g r a l (a - x^{2})^{\frac{1}{2}}

integral sin^3(x)cos^3(x)

in t e g r a l sin^{3} (x) cos^{3} (x)