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

Lösung

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

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 4 + arctan (\frac{x}{2})) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x + 2}{x ( x ^{2} + 4 )} : \frac{1}{2 x} + \frac{- x + 2}{2 ( x ^{2} + 4 )}

= \int \frac{1}{2 x} + \frac{- x + 2}{2 ( 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{1}{2 x} d x + \int \frac{- x + 2}{2 ( x ^{2} + 4 )} d x

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

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

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 4 + arctan (\frac{x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} (- \frac{1}{2} ln x^{2} + 4 + arctan (\frac{x}{2})) + C

\frac{d}{d x} (2 sec^{2} (5 π x - 1))

\frac{d}{d x} (\frac{5}{x} + \frac{x}{5})

\frac{d y}{d x} = 16 x + y - 16

\int x ln (\frac{1}{x}) d x

s l o p e (1.1) (2.4)