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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{\partial}{\partial y} ((x + y) e^{y})

x \frac{d y}{d x} - 3 y = x^{4}

\int_{1}^{e} ln (5 x^{3}) d x

\int_{0}^{5} (7 x - x^{2}) - 2 x d x

\frac{\partial}{\partial y} (e^{x y^{2}})