integral of ((2x+2))/(x^2+10x+24)

Lösung

\int \frac{( 2 x + 2 )}{x ^{2} + 10 x + 24} d x

ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣ + C

Schritte zur Lösung

\int \frac{2 x + 2}{x ^{2} + 10 x + 24} d x

Multipliziere aus

\frac{2 x + 2}{x ^{2} + 10 x + 24} : \frac{2 x}{x ^{2} + 10 x + 24} + \frac{2}{x ^{2} + 10 x + 24}

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{2 x}{x ^{2} + 10 x + 24} d x + \int \frac{2}{x ^{2} + 10 x + 24} d x

\int \frac{2 x}{x ^{2} + 10 x + 24} d x = ln x^{2} + 10 x + 24 + 5 ln ∣ x + 6 ∣ - 5 ln ∣ x + 4 ∣

\int \frac{2}{x ^{2} + 10 x + 24} d x = ln ∣ x + 4 ∣ - ln ∣ x + 6 ∣

= ln x^{2} + 10 x + 24 + 5 ln ∣ x + 6 ∣ - 5 ln ∣ x + 4 ∣ + ln ∣ x + 4 ∣ - ln ∣ x + 6 ∣

Vereinfache

= ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣ + C

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