integral of (x^2+2)/(x^3+6x)

Lösung

\int \frac{x ^{2} + 2}{x ^{3} + 6 x} d x

\frac{1}{3} ln x^{2} + 6 + \frac{1}{3} ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{x ^{2} + 2}{x ^{3} + 6 x} d x

Multipliziere aus

\frac{x ^{2} + 2}{x ^{3} + 6 x} : \frac{x ^{2}}{x ^{3} + 6 x} + \frac{2}{x ^{3} + 6 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 ^{2}}{x ^{3} + 6 x} d x + \int \frac{2}{x ^{3} + 6 x} d x

\int \frac{x ^{2}}{x ^{3} + 6 x} d x = \frac{1}{2} ln x^{2} + 6

\int \frac{2}{x ^{3} + 6 x} d x = 2 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{12} ln x^{2} + 6)

= \frac{1}{2} ln x^{2} + 6 + 2 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{12} ln x^{2} + 6)

Vereinfache

\frac{1}{2} ln x^{2} + 6 + 2 (\frac{1}{6} ln ∣ x ∣ - \frac{1}{12} ln x^{2} + 6) : \frac{1}{3} ln x^{2} + 6 + \frac{1}{3} ln ∣ x ∣

= \frac{1}{3} ln x^{2} + 6 + \frac{1}{3} ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln x^{2} + 6 + \frac{1}{3} ln ∣ x ∣ + C

d er i v a t i v e sec (\frac{π}{3}) tan (\frac{π}{3})

(11 x^{2} \cdot y - 13 x) d y = - (11 x y^{2} - 13 y) d x

\int_{0}^{1} (x^{2} + 3) e^{- x} d x

\int - y e^{y} d y

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