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Beliebt Rechnen >

integral of (5x^2-7x+10)/(x^3+12x^2+36x)

Lösung

\int \frac{5 x ^{2} - 7 x + 10}{x ^{3} + 12 x ^{2} + 36 x} d x

Lösung

\frac{5}{18} ln ∣ x ∣ + \frac{85}{18} ln ∣ x + 6 ∣ + \frac{116}{3 ( x + 6 )} + C

Schritte zur Lösung

\int \frac{5 x ^{2} - 7 x + 10}{x ^{3} + 12 x ^{2} + 36 x} d x

Ermittle den Partialbruch von

\frac{5 x ^{2} - 7 x + 10}{x ^{3} + 12 x ^{2} + 36 x} : \frac{5}{18 x} + \frac{85}{18 ( x + 6 )} - \frac{116}{3 ( x + 6 ) ^{2}}

= \int \frac{5}{18 x} + \frac{85}{18 ( x + 6 )} - \frac{116}{3 ( x + 6 ) ^{2}} 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{5}{18 x} d x + \int \frac{85}{18 ( x + 6 )} d x - \int \frac{116}{3 ( x + 6 ) ^{2}} d x

\int \frac{5}{18 x} d x = \frac{5}{18} ln ∣ x ∣

\int \frac{85}{18 ( x + 6 )} d x = \frac{85}{18} ln ∣ x + 6 ∣

\int \frac{116}{3 ( x + 6 ) ^{2}} d x = - \frac{116}{3 ( x + 6 )}

= \frac{5}{18} ln ∣ x ∣ + \frac{85}{18} ln ∣ x + 6 ∣ - (- \frac{116}{3 ( x + 6 )})

Vereinfache

= \frac{5}{18} ln ∣ x ∣ + \frac{85}{18} ln ∣ x + 6 ∣ + \frac{116}{3 ( x + 6 )}

Füge eine Konstante zur Lösung hinzu

= \frac{5}{18} ln ∣ x ∣ + \frac{85}{18} ln ∣ x + 6 ∣ + \frac{116}{3 ( x + 6 )} + C

Graph

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