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Beliebt Rechnen >

integral of (x^4-6)/(x^3-2x^2)

Lösung

\int \frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}} d x

Lösung

\frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}} d x

Faktorisiere

\frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}}

= \int \frac{( x + 4 6 ) ( x - 4 6 ) ( x ^{2} + 6 )}{x ^{2} ( x - 2 )} d x

Ermittle den Partialbruch von

\frac{( x + 4 6 ) ( x - 4 6 ) ( x ^{2} + 6 )}{x ^{2} ( x - 2 )} : x + 2 + \frac{3}{2 x} + \frac{3}{x ^{2}} + \frac{5}{2 ( x - 2 )}

= \int x + 2 + \frac{3}{2 x} + \frac{3}{x ^{2}} + \frac{5}{2 ( x - 2 )} d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x d x + \int 2 d x + \int \frac{3}{2 x} d x + \int \frac{3}{x ^{2}} d x + \int \frac{5}{2 ( x - 2 )} d x

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

\int 2 d x = 2 x

\int \frac{3}{2 x} d x = \frac{3}{2} ln ∣ x ∣

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

\int \frac{5}{2 ( x - 2 )} d x = \frac{5}{2} ln ∣ x - 2 ∣

= \frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of (x^a-x^{-a})^2

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

integral of x/(sqrt(25+3x))

\int \frac{x}{25 + 3 x} d x

integral of ,^3x

integral of 1/(sqrt(-4x^2-12x-8))

\int \frac{1}{- 4 x ^{2} - 12 x - 8} d x

integral of (12x^2-4x)/((x^2+4)(x+2))

\int \frac{12 x ^{2} - 4 x}{( x ^{2} + 4 ) ( x + 2 )} d x