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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

\int \frac{2}{x - 2} d x = 2 ln ∣ x - 2 ∣

= 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 - 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + 2 ln ∣ x - 2 ∣

Vereinfache

3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 - 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + 2 ln ∣ x - 2 ∣ : 3 x^{2} - 12 x + \frac{1}{3} (x - 2)^{3} - 2 ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 12

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 3/(t(1+t^2))

\int \frac{3}{t ( 1 + t ^{2} )} d t

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

\int \frac{4 x ^{2} - x}{x} d x

integral of 1/(g-kv^2)

\int \frac{1}{g - k v ^{2}} d v

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

integral of (\partial)/(\partial x)(x^2)

\int \frac{\partial}{\partial x} (x^{2}) d x