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Beliebt Rechnen >

integral of (3x+1)/((x^2+x-6)(x+2))

Lösung

\int \frac{3 x + 1}{( x ^{2} + x - 6 ) ( x + 2 )} d x

Lösung

\frac{7}{20} ln ∣ x - 2 ∣ - \frac{8}{5} ln ∣ x + 3 ∣ + \frac{5}{4} ln ∣ x + 2 ∣ + C

Schritte zur Lösung

\int \frac{3 x + 1}{( x ^{2} + x - 6 ) ( x + 2 )} d x

Ermittle den Partialbruch von

\frac{3 x + 1}{( x ^{2} + x - 6 ) ( x + 2 )} : \frac{7}{20 ( x - 2 )} - \frac{8}{5 ( x + 3 )} + \frac{5}{4 ( x + 2 )}

= \int \frac{7}{20 ( x - 2 )} - \frac{8}{5 ( x + 3 )} + \frac{5}{4 ( 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 \frac{7}{20 ( x - 2 )} d x - \int \frac{8}{5 ( x + 3 )} d x + \int \frac{5}{4 ( x + 2 )} d x

\int \frac{7}{20 ( x - 2 )} d x = \frac{7}{20} ln ∣ x - 2 ∣

\int \frac{8}{5 ( x + 3 )} d x = \frac{8}{5} ln ∣ x + 3 ∣

\int \frac{5}{4 ( x + 2 )} d x = \frac{5}{4} ln ∣ x + 2 ∣

= \frac{7}{20} ln ∣ x - 2 ∣ - \frac{8}{5} ln ∣ x + 3 ∣ + \frac{5}{4} ln ∣ x + 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{7}{20} ln ∣ x - 2 ∣ - \frac{8}{5} ln ∣ x + 3 ∣ + \frac{5}{4} ln ∣ x + 2 ∣ + C

Graph

Beliebte Beispiele

integral of (sqrt(x)+1)^{1/4}

\int (x + 1)^{\frac{1}{4}} d x

integral of x^2(4+x^3)

\int x^{2} (4 + x^{3}) d x

integral of 8x(1-x)

\int 8 x (1 - x) d x

integral of e^tcot(e^t-8)

\int e^{t} cot (e^{t} - 8) d t

integral of (x^3)/((x^4+9)^4)

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