integral of (x-1)/(x^3+x)

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

f (x) = x e^{4 x}

\int 4 csc (x) cot (x) + 2 sec^{2} (x) d x

\frac{\partial}{\partial y} (5 e^{x} ln (y))

\frac{d}{d x} (e^{x} (sin (x) + cos (x)))

t a y l or x^{3} - 2 x^{2} + 3 x + 5 (2)