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Beliebt Rechnen >

integral of 6/(x^2+10x+16)

Lösung

\int \frac{6}{x ^{2} + 10 x + 16} d x

Lösung

ln ∣ x + 2 ∣ - ln ∣ x + 8 ∣ + C

Schritte zur Lösung

\int \frac{6}{x ^{2} + 10 x + 16} d x

Entferne die Konstante:

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

= 6 \cdot \int \frac{1}{x ^{2} + 10 x + 16} d x

Ermittle den Partialbruch von

\frac{1}{x ^{2} + 10 x + 16} : \frac{1}{6 ( x + 2 )} - \frac{1}{6 ( x + 8 )}

= 6 \cdot \int \frac{1}{6 ( x + 2 )} - \frac{1}{6 ( x + 8 )} 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

= 6 (\int \frac{1}{6 ( x + 2 )} d x - \int \frac{1}{6 ( x + 8 )} d x)

\int \frac{1}{6 ( x + 2 )} d x = \frac{1}{6} ln ∣ x + 2 ∣

\int \frac{1}{6 ( x + 8 )} d x = \frac{1}{6} ln ∣ x + 8 ∣

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

Vereinfache

6 (\frac{1}{6} ln ∣ x + 2 ∣ - \frac{1}{6} ln ∣ x + 8 ∣) : ln ∣ x + 2 ∣ - ln ∣ x + 8 ∣

= ln ∣ x + 2 ∣ - ln ∣ x + 8 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x + 2 ∣ - ln ∣ x + 8 ∣ + C

Graph

Beliebte Beispiele

integral of 6sin^4(x)

\int 6 sin^{4} (x) d x

derivative y=ln((x^2-1)/(x^2+1))

d er i v a t i v e y = ln (\frac{x ^{2} - 1}{x ^{2} + 1})

derivative-10e^{-5x^2}x

d er i v a t i v e - 10 e^{- 5 x^{2}} x

integral of 6xln(5x)

\int 6 x ln (5 x) d x

fläche y=x^4-4x^2+4,y=x^2

a re a y = x^{4} - 4 x^{2} + 4, y = x^{2}