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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{10} ln ∣ x - 2 ∣ - \frac{3}{5} ln ∣ x + 3 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x}{( x ^{2} - 4 ) ( x + 3 )} : \frac{1}{2 ( x + 2 )} + \frac{1}{10 ( x - 2 )} - \frac{3}{5 ( x + 3 )}

= \int \frac{1}{2 ( x + 2 )} + \frac{1}{10 ( x - 2 )} - \frac{3}{5 ( x + 3 )} 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{1}{2 ( x + 2 )} d x + \int \frac{1}{10 ( x - 2 )} d x - \int \frac{3}{5 ( x + 3 )} d x

\int \frac{1}{2 ( x + 2 )} d x = \frac{1}{2} ln ∣ x + 2 ∣

\int \frac{1}{10 ( x - 2 )} d x = \frac{1}{10} ln ∣ x - 2 ∣

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

= \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{10} ln ∣ x - 2 ∣ - \frac{3}{5} ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{10} ln ∣ x - 2 ∣ - \frac{3}{5} ln ∣ x + 3 ∣ + C

Graph

Beliebte Beispiele

integral of 15^x

\int 1 5^{x} d x

tangent f(x)= 5/(2sqrt(5x+6)),\at x=2

t an g e n t f (x) = \frac{5}{2 5 x + 6}, at x = 2

integral of ze^z

\int z e^{z} d z

derivative y=sqrt(10+cot^2(x))

d er i v a t i v e y = 10 + cot^{2} (x)

derivative of (8e^x/(2e^x+1))

\frac{d}{d x} (\frac{8 e ^{x}}{2 e ^{x} + 1})