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Beliebt Rechnen >

integral of (x^2+2x+3)/((x-6)(x^2+4))

Lösung

\int \frac{x ^{2} + 2 x + 3}{( x - 6 ) ( x ^{2} + 4 )} d x

Lösung

\frac{51}{40} ln ∣ x - 6 ∣ + \frac{1}{40} (- \frac{11}{2} ln x^{2} + 4 + 7 arctan (\frac{x}{2})) + C

Schritte zur Lösung

\int \frac{x ^{2} + 2 x + 3}{( x - 6 ) ( x ^{2} + 4 )} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 2 x + 3}{( x - 6 ) ( x ^{2} + 4 )} : \frac{51}{40 ( x - 6 )} + \frac{- 11 x + 14}{40 ( x ^{2} + 4 )}

= \int \frac{51}{40 ( x - 6 )} + \frac{- 11 x + 14}{40 ( x ^{2} + 4 )} 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{51}{40 ( x - 6 )} d x + \int \frac{- 11 x + 14}{40 ( x ^{2} + 4 )} d x

\int \frac{51}{40 ( x - 6 )} d x = \frac{51}{40} ln ∣ x - 6 ∣

\int \frac{- 11 x + 14}{40 ( x ^{2} + 4 )} d x = \frac{1}{40} (- \frac{11}{2} ln x^{2} + 4 + 7 arctan (\frac{x}{2}))

= \frac{51}{40} ln ∣ x - 6 ∣ + \frac{1}{40} (- \frac{11}{2} ln x^{2} + 4 + 7 arctan (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{51}{40} ln ∣ x - 6 ∣ + \frac{1}{40} (- \frac{11}{2} ln x^{2} + 4 + 7 arctan (\frac{x}{2})) + C

Graph

Beliebte Beispiele

integral of sqrt(12+8x-3x^2)

\int 12 + 8 x - 3 x^{2} d x

integral of (3t^2-sqrt(5t)+2)

\int (3 t^{2} - 5 t + 2) d t

integral of 5/(sqrt(x^2+4))

\int \frac{5}{x ^{2} + 4} d x

integral of (x+3)/(x^2+4x+13)

\int \frac{x + 3}{x ^{2} + 4 x + 13} d x

integral of 1/(4x^2+12x+25)

\int \frac{1}{4 x ^{2} + 12 x + 25} d x