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Beliebt Rechnen >

integral of (2x+18)/(x^2+10x+24)

Lösung

\int \frac{2 x + 18}{x ^{2} + 10 x + 24} d x

Lösung

ln x^{2} + 10 x + 24 - 8 (\frac{1}{2} ln ∣ x + 6 ∣ - \frac{1}{2} ln ∣ x + 4 ∣) + C

Schritte zur Lösung

\int \frac{2 x + 18}{x ^{2} + 10 x + 24} d x

Vervollständige das Quadrat

x^{2} + 10 x + 24 : (x + 5)^{2} - 1

= \int \frac{2 x + 18}{( x + 5 ) ^{2} - 1} d x

Wende U-Substitution an

= \int \frac{2 u + 8}{u ^{2} - 1} d u

Multipliziere aus

\frac{2 u + 8}{u ^{2} - 1} : \frac{2 u}{u ^{2} - 1} + \frac{8}{u ^{2} - 1}

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 u}{u ^{2} - 1} d u + \int \frac{8}{u ^{2} - 1} d u

\int \frac{2 u}{u ^{2} - 1} d u = ln u^{2} - 1

\int \frac{8}{u ^{2} - 1} d u = - 8 (\frac{1}{2} ln ∣ u + 1 ∣ - \frac{1}{2} ln ∣ u - 1 ∣)

= ln u^{2} - 1 - 8 (\frac{1}{2} ln ∣ u + 1 ∣ - \frac{1}{2} ln ∣ u - 1 ∣)

Setze in

u = x + 5

ein

= ln (x + 5)^{2} - 1 - 8 (\frac{1}{2} ln ∣ x + 5 + 1 ∣ - \frac{1}{2} ln ∣ x + 5 - 1 ∣)

Vereinfache

ln (x + 5)^{2} - 1 - 8 (\frac{1}{2} ln ∣ x + 5 + 1 ∣ - \frac{1}{2} ln ∣ x + 5 - 1 ∣) : ln x^{2} + 10 x + 24 - 8 (\frac{1}{2} ln ∣ x + 6 ∣ - \frac{1}{2} ln ∣ x + 4 ∣)

= ln x^{2} + 10 x + 24 - 8 (\frac{1}{2} ln ∣ x + 6 ∣ - \frac{1}{2} ln ∣ x + 4 ∣)

Füge eine Konstante zur Lösung hinzu

= ln x^{2} + 10 x + 24 - 8 (\frac{1}{2} ln ∣ x + 6 ∣ - \frac{1}{2} ln ∣ x + 4 ∣) + C

Graph

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