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Beliebt Rechnen >

integral of (3x-9)/((x+4)(x-5))

Lösung

\int \frac{3 x - 9}{( x + 4 ) ( x - 5 )} d x

Lösung

3 ln ∣ x + 4 ∣ + 6 (\frac{1}{9} ln ∣ x - 5 ∣ - \frac{1}{9} ln ∣ x + 4 ∣) + C

Schritte zur Lösung

\int \frac{3 x - 9}{( x + 4 ) ( x - 5 )} d x

Wende U-Substitution an

= \int \frac{3 u + 6}{u ( u + 9 )} d u

Multipliziere aus

\frac{3 u + 6}{u ( u + 9 )} : \frac{3 u}{u ( u + 9 )} + \frac{6}{u ( u + 9 )}

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{3 u}{u ( u + 9 )} d u + \int \frac{6}{u ( u + 9 )} d u

\int \frac{3 u}{u ( u + 9 )} d u = 3 ln ∣ u + 9 ∣

\int \frac{6}{u ( u + 9 )} d u = 6 (\frac{1}{9} ln ∣ u ∣ - \frac{1}{9} ln ∣ u + 9 ∣)

= 3 ln ∣ u + 9 ∣ + 6 (\frac{1}{9} ln ∣ u ∣ - \frac{1}{9} ln ∣ u + 9 ∣)

Setze in

u = x - 5

ein

= 3 ln ∣ x - 5 + 9 ∣ + 6 (\frac{1}{9} ln ∣ x - 5 ∣ - \frac{1}{9} ln ∣ x - 5 + 9 ∣)

Vereinfache

3 ln ∣ x - 5 + 9 ∣ + 6 (\frac{1}{9} ln ∣ x - 5 ∣ - \frac{1}{9} ln ∣ x - 5 + 9 ∣) : 3 ln ∣ x + 4 ∣ + 6 (\frac{1}{9} ln ∣ x - 5 ∣ - \frac{1}{9} ln ∣ x + 4 ∣)

= 3 ln ∣ x + 4 ∣ + 6 (\frac{1}{9} ln ∣ x - 5 ∣ - \frac{1}{9} ln ∣ x + 4 ∣)

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x + 4 ∣ + 6 (\frac{1}{9} ln ∣ x - 5 ∣ - \frac{1}{9} ln ∣ x + 4 ∣) + C

Graph

Beliebte Beispiele

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

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

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

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

integral of (sin^3(y))/(1+cos(y))

\int \frac{sin ^{3} ( y )}{1 + cos ( y )} d y

integral of (cot^2(t))/(csc(t))

\int \frac{cot ^{2} ( t )}{csc ( t )} d t

integral of (x^2-4)/(x^2-16)

\int \frac{x ^{2} - 4}{x ^{2} - 16} d x