integral of (x^2+3x+1)/(x^4+5x^2+4)

Lösung

\int \frac{x ^{2} + 3 x + 1}{x ^{4} + 5 x ^{2} + 4} d x

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

Schritte zur Lösung

\int \frac{x ^{2} + 3 x + 1}{x ^{4} + 5 x ^{2} + 4} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 3 x + 1}{x ^{4} + 5 x ^{2} + 4} : \frac{x}{x ^{2} + 1} + \frac{- x + 1}{x ^{2} + 4}

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} ((3 x^{2} - 10)^{- 10})

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

x \to - 2 lim (x^{3})

d er i v a t i v e \frac{3 + 8 x}{9 - 2 x}

d er i v a t i v e f (t) = (1 t^{2} + 0 t + 0)^{- 8}