integral of 1/(x+x^3)

Lösung

\int \frac{1}{x + x ^{3}} d x

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

Schritte zur Lösung

\int \frac{1}{x + x ^{3}} d x

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

Füge eine Konstante zur Lösung hinzu

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

\frac{\partial}{\partial x} (e^{- \frac{x ^{2}}{6} - \frac{y ^{2}}{6}})

n = 1 \sum \infty (cos (1.6))^{n}

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

x \to 0 - lim (\frac{∣ 8 x ∣}{x})

\frac{d}{d x} (x \cdot e^{- b x^{2}})