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Beliebt Rechnen >

integral of 7/((x^2+3x-10))

Lösung

\int \frac{7}{( x ^{2} + 3 x - 10 )} d x

Lösung

ln ∣ x - 2 ∣ - ln ∣ x + 5 ∣ + C

Schritte zur Lösung

\int \frac{7}{x ^{2} + 3 x - 10} d x

Entferne die Konstante:

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

= 7 \cdot \int \frac{1}{x ^{2} + 3 x - 10} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} + 3 x - 10} : \frac{1}{7 ( x - 2 )} - \frac{1}{7 ( x + 5 )}

= 7 \cdot \int \frac{1}{7 ( x - 2 )} - \frac{1}{7 ( x + 5 )} 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

= 7 (\int \frac{1}{7 ( x - 2 )} d x - \int \frac{1}{7 ( x + 5 )} d x)

\int \frac{1}{7 ( x - 2 )} d x = \frac{1}{7} ln ∣ x - 2 ∣

\int \frac{1}{7 ( x + 5 )} d x = \frac{1}{7} ln ∣ x + 5 ∣

= 7 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣)

Vereinfache

7 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣) : ln ∣ x - 2 ∣ - ln ∣ x + 5 ∣

= ln ∣ x - 2 ∣ - ln ∣ x + 5 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 2 ∣ - ln ∣ x + 5 ∣ + C

Graph

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