integral of (4x^2+x+27)/(x^3+9x)

Lösung

\int \frac{4 x ^{2} + x + 27}{x ^{3} + 9 x} d x

3 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int \frac{4 x ^{2} + x + 27}{x ^{3} + 9 x} d x

Ermittle den Partialbruch von

\frac{4 x ^{2} + x + 27}{x ^{3} + 9 x} : \frac{3}{x} + \frac{x + 1}{x ^{2} + 9}

= \int \frac{3}{x} + \frac{x + 1}{x ^{2} + 9} 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}{x} d x + \int \frac{x + 1}{x ^{2} + 9} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

\int \frac{x + 1}{x ^{2} + 9} d x = \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})

= 3 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= 3 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3}) + C

\frac{\partial}{\partial x} (2 x^{2} - 7 y^{9} - 4)

e^{x} sin (x) + e^{x} cos (x)

\int y y^{2} + 1 d y

x \to 3 lim (2 x + 4)