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Beliebt Rechnen >

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

Lösung

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

Lösung

x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3 + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int \frac{6}{x ( x - 3 )} d x = 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣)

= x - 3 + 3 ln ∣ x - 3 ∣ - 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣)

Vereinfache

x - 3 + 3 ln ∣ x - 3 ∣ - 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣) : x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3

= x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3

Füge eine Konstante zur Lösung hinzu

= x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3 + C

Graph

Beliebte Beispiele

integral of (8x)/(sqrt(2x^2+1))

\int \frac{8 x}{2 x ^{2} + 1} d x

integral of (7x^2-9x+6)/((x-1)(x^2+1))

\int \frac{7 x ^{2} - 9 x + 6}{( x - 1 ) ( x ^{2} + 1 )} d x

integral of 19xsin(x)cos(x)

\int 19 x sin (x) cos (x) d x

integral of xex^2-1

\int x e x^{2} - 1 d x

integral of sin^4(7θ)

\int sin^{4} (7 θ) d θ