integral of (3x^5+2)/(-5x^5+5x)

Lösung

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

- \frac{3}{5} x + \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- ln x^{2} + 1 + 3 arctan (x)) + \frac{1}{20} ln ∣ x + 1 ∣ - \frac{1}{4} ln ∣ x - 1 ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{3 x ^{5} + 2}{- 5 x ^{5} + 5 x} : - \frac{3}{5} + \frac{2}{5 x} + \frac{- 2 x + 3}{10 ( x ^{2} + 1 )} + \frac{1}{20 ( x + 1 )} - \frac{1}{4 ( x - 1 )}

= \int - \frac{3}{5} + \frac{2}{5 x} + \frac{- 2 x + 3}{10 ( x ^{2} + 1 )} + \frac{1}{20 ( x + 1 )} - \frac{1}{4 ( x - 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{3}{5} d x + \int \frac{2}{5 x} d x + \int \frac{- 2 x + 3}{10 ( x ^{2} + 1 )} d x + \int \frac{1}{20 ( x + 1 )} d x - \int \frac{1}{4 ( x - 1 )} d x

\int \frac{3}{5} d x = \frac{3}{5} x

\int \frac{2}{5 x} d x = \frac{2}{5} ln ∣ x ∣

\int \frac{- 2 x + 3}{10 ( x ^{2} + 1 )} d x = \frac{1}{10} (- ln x^{2} + 1 + 3 arctan (x))

\int \frac{1}{20 ( x + 1 )} d x = \frac{1}{20} ln ∣ x + 1 ∣

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

= - \frac{3}{5} x + \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- ln x^{2} + 1 + 3 arctan (x)) + \frac{1}{20} ln ∣ x + 1 ∣ - \frac{1}{4} ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{5} x + \frac{2}{5} ln ∣ x ∣ + \frac{1}{10} (- ln x^{2} + 1 + 3 arctan (x)) + \frac{1}{20} ln ∣ x + 1 ∣ - \frac{1}{4} ln ∣ x - 1 ∣ + C

\frac{d}{d x} (4 e^{x})

\int 2 e^{3} d x

\frac{d}{d x} (a \cdot sinh (\frac{x}{a}))

\int \frac{1}{4 x ^{2} + 16} d x

\frac{d}{d t} (t cos (2 t))