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Beliebt Rechnen >

integral of 9/(x^2+3x-18)

Lösung

\int \frac{9}{x ^{2} + 3 x - 18} d x

Lösung

ln ∣ x - 3 ∣ - ln ∣ x + 6 ∣ + C

Schritte zur Lösung

\int \frac{9}{x ^{2} + 3 x - 18} d x

Entferne die Konstante:

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

= 9 \cdot \int \frac{1}{x ^{2} + 3 x - 18} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} + 3 x - 18} : \frac{1}{9 ( x - 3 )} - \frac{1}{9 ( x + 6 )}

= 9 \cdot \int \frac{1}{9 ( x - 3 )} - \frac{1}{9 ( x + 6 )} 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

= 9 (\int \frac{1}{9 ( x - 3 )} d x - \int \frac{1}{9 ( x + 6 )} d x)

\int \frac{1}{9 ( x - 3 )} d x = \frac{1}{9} ln ∣ x - 3 ∣

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

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

Vereinfache

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

= ln ∣ x - 3 ∣ - ln ∣ x + 6 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 3 ∣ - ln ∣ x + 6 ∣ + C

Graph

Beliebte Beispiele

integral from-3 to 1 of |3x+3|

\int_{- 3}^{1} ∣ 3 x + 3 ∣ d x

x^2y^'=2y^2-x^2

x^{2} y^{^{'}} = 2 y^{2} - x^{2}

integral of sinh^2(2x)

\int sinh^{2} (2 x) d x

integral of 7/(xln(x))

\int \frac{7}{x ln ( x )} d x

(\partial)/(\partial x)((x-4)^2+(y-4)^2-5)

\frac{\partial}{\partial x} ((x - 4)^{2} + (y - 4)^{2} - 5)