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Beliebt Rechnen >

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

Lösung

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

Lösung

ln ∣ x - 9 ∣ - ln ∣ x - 4 ∣ + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{( x - 9 ) ( x - 4 )} d x

Ermittle den Partialbruch von

\frac{1}{( x - 9 ) ( x - 4 )} : \frac{1}{5 ( x - 9 )} - \frac{1}{5 ( x - 4 )}

= 5 \cdot \int \frac{1}{5 ( x - 9 )} - \frac{1}{5 ( 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

= 5 (\int \frac{1}{5 ( x - 9 )} d x - \int \frac{1}{5 ( x - 4 )} d x)

\int \frac{1}{5 ( x - 9 )} d x = \frac{1}{5} ln ∣ x - 9 ∣

\int \frac{1}{5 ( x - 4 )} d x = \frac{1}{5} ln ∣ x - 4 ∣

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

Vereinfache

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

= ln ∣ x - 9 ∣ - ln ∣ x - 4 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 9 ∣ - ln ∣ x - 4 ∣ + C

Graph

Beliebte Beispiele

integral of 2cos^2(t)

\int 2 cos^{2} (t) d t

integral of 2/(4-2x)

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

integral of 1/pi xsin(x)

\int \frac{1}{π} x sin (x) d x

integral of x(9x^2-5)^{3/2}

\int x (9 x^{2} - 5)^{\frac{3}{2}} d x

integral of sin(5x+2)

\int sin (5 x + 2) d x