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

Lösung

integral

\frac{( x + 2 )}{( x ^{2} + 3 x )}

\frac{2}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x + 3 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{x}{x ^{2} + 3 x} d x = ln ∣ x + 3 ∣

\int \frac{2}{x ^{2} + 3 x} d x = 2 (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 3 ∣)

= ln ∣ x + 3 ∣ + 2 (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 3 ∣)

Vereinfache

ln ∣ x + 3 ∣ + 2 (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x + 3 ∣) : \frac{2}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x + 3 ∣

= \frac{2}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x + 3 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x + 3 ∣ + C

in t e g r a l \frac{v}{2}

in t e g r a l \frac{( x - 1 )}{( x ^{2} + 2 x + 3 )}

in t e g r a l \frac{1}{x ( x ^{2} - 1 )}

in t e g r a l \frac{x ^{- 2} ( e ^{x} ) ^{3}}{3}

in t e g r a l \frac{1}{( 3 x ^{2} + 1 )}