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Beliebt Rechnen >

integral of (((x+2))/(x^3+2x^2+5x))

Lösung

\int (\frac{( x + 2 )}{x ^{3} + 2 x ^{2} + 5 x}) d x

Lösung

\frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- 2 ln x^{2} + 2 x + 5 + 3 arctan (\frac{1}{2} (x + 1))) + C

Schritte zur Lösung

\int \frac{x + 2}{x ^{3} + 2 x ^{2} + 5 x} d x

Ermittle den Partialbruch von

\frac{x + 2}{x ^{3} + 2 x ^{2} + 5 x} : \frac{2}{5 x} + \frac{- 2 x + 1}{5 ( x ^{2} + 2 x + 5 )}

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

\int \frac{2}{5 x} d x = \frac{2}{5} ln ∣ x ∣

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

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

Vereinfache

\frac{2}{5} ln ∣ x ∣ + \frac{1}{5} (- ln x^{2} + 2 x + 5 + arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} arctan (\frac{1}{2} (x + 1))) : \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- 2 ln x^{2} + 2 x + 5 + 3 arctan (\frac{1}{2} (x + 1)))

= \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- 2 ln x^{2} + 2 x + 5 + 3 arctan (\frac{1}{2} (x + 1)))

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- 2 ln x^{2} + 2 x + 5 + 3 arctan (\frac{1}{2} (x + 1))) + C

Graph

Beliebte Beispiele

integral of cot^2(z)

\int cot^{2} (z) d z

integral of ((2-x))/x

\int \frac{( 2 - x )}{x} d x

integral of ((-1)^x)/x

\int \frac{( - 1 ) ^{x}}{x} d x

integral of (e^{2x})/(e^{2x)-4}

\int \frac{e ^{2 x}}{e ^{2 x} - 4} d x

integral of (cos(x))(sin(x))^2

\int (cos (x)) (sin (x))^{2} d x