integral of 1/((x-2)(x-3)(x-4))

Lösung

\int \frac{1}{( x - 2 ) ( x - 3 ) ( x - 4 )} d x

\frac{1}{2} ln ∣ x - 2 ∣ - ln ∣ x - 3 ∣ + \frac{1}{2} ln ∣ x - 4 ∣ + C

Schritte zur Lösung

\int \frac{1}{( x - 2 ) ( x - 3 ) ( x - 4 )} d x

Ermittle den Partialbruch von

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

= \int \frac{1}{2 ( x - 2 )} - \frac{1}{x - 3} + \frac{1}{2 ( x - 4 )} 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}{x - 3} d x + \int \frac{1}{2 ( x - 4 )} d x

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

\int \frac{1}{x - 3} d x = ln ∣ x - 3 ∣

\int \frac{1}{2 ( x - 4 )} d x = \frac{1}{2} ln ∣ x - 4 ∣

= \frac{1}{2} ln ∣ x - 2 ∣ - ln ∣ x - 3 ∣ + \frac{1}{2} ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x - 2 ∣ - ln ∣ x - 3 ∣ + \frac{1}{2} ln ∣ x - 4 ∣ + C

x \to 2 lim (\frac{5}{( x - 2 ) ^{2}})

\frac{\partial}{\partial s} (\frac{s - 3 r}{t})

\frac{\partial}{\partial x} (y (1 - x^{2}))

x \to 2 + lim (9 - x^{2})

\int_{1}^{2} ln (3) x^{2} d x