integral of (5x^2-3x+18)/(9x-x^3)

Lösung

\int \frac{5 x ^{2} - 3 x + 18}{9 x - x ^{3}} d x

- \frac{5}{2} ln - x^{2} + 9 - \frac{3}{2} ln ∣ x + 3 ∣ - \frac{1}{2} ln ∣ x - 3 ∣ - \frac{3 x}{- x ^{2} + 9} - \frac{3}{2 ( x + 3 )} - \frac{3}{2 ( x - 3 )} + 2 ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{5 x ^{2} - 3 x + 18}{9 x - x ^{3}} d x

Multipliziere aus

\frac{5 x ^{2} - 3 x + 18}{9 x - x ^{3}} : \frac{5 x ^{2}}{9 x - x ^{3}} - \frac{3 x}{9 x - x ^{3}} + \frac{18}{9 x - x ^{3}}

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{5 x ^{2}}{9 x - x ^{3}} d x - \int \frac{3 x}{9 x - x ^{3}} d x + \int \frac{18}{9 x - x ^{3}} d x

\int \frac{5 x ^{2}}{9 x - x ^{3}} d x = - \frac{5}{2} ln 9 - x^{2}

\int \frac{3 x}{9 x - x ^{3}} d x = 3 (\frac{x}{9 - x ^{2}} - 2 (- \frac{1}{12} ln ∣ x + 3 ∣ - \frac{1}{4 ( x + 3 )} + \frac{1}{12} ln ∣ x - 3 ∣ - \frac{1}{4 ( x - 3 )}))

\int \frac{18}{9 x - x ^{3}} d x = 2 ln ∣ x ∣ - ln ∣ x + 3 ∣ - ln ∣ x - 3 ∣

= - \frac{5}{2} ln 9 - x^{2} - 3 (\frac{x}{9 - x ^{2}} - 2 (- \frac{1}{12} ln ∣ x + 3 ∣ - \frac{1}{4 ( x + 3 )} + \frac{1}{12} ln ∣ x - 3 ∣ - \frac{1}{4 ( x - 3 )})) + 2 ln ∣ x ∣ - ln ∣ x + 3 ∣ - ln ∣ x - 3 ∣

Vereinfache

- \frac{5}{2} ln 9 - x^{2} - 3 (\frac{x}{9 - x ^{2}} - 2 (- \frac{1}{12} ln ∣ x + 3 ∣ - \frac{1}{4 ( x + 3 )} + \frac{1}{12} ln ∣ x - 3 ∣ - \frac{1}{4 ( x - 3 )})) + 2 ln ∣ x ∣ - ln ∣ x + 3 ∣ - ln ∣ x - 3 ∣ : - \frac{5}{2} ln - x^{2} + 9 - \frac{3}{2} ln ∣ x + 3 ∣ - \frac{1}{2} ln ∣ x - 3 ∣ - \frac{3 x}{- x ^{2} + 9} - \frac{3}{2 ( x + 3 )} - \frac{3}{2 ( x - 3 )} + 2 ln ∣ x ∣

= - \frac{5}{2} ln - x^{2} + 9 - \frac{3}{2} ln ∣ x + 3 ∣ - \frac{1}{2} ln ∣ x - 3 ∣ - \frac{3 x}{- x ^{2} + 9} - \frac{3}{2 ( x + 3 )} - \frac{3}{2 ( x - 3 )} + 2 ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{2} ln - x^{2} + 9 - \frac{3}{2} ln ∣ x + 3 ∣ - \frac{1}{2} ln ∣ x - 3 ∣ - \frac{3 x}{- x ^{2} + 9} - \frac{3}{2 ( x + 3 )} - \frac{3}{2 ( x - 3 )} + 2 ln ∣ x ∣ + C

\frac{d}{d x} (e^{a x} + arctan (c x))

d er i v a t i v e y = \frac{7}{3 x ^{4}}

\int \frac{x ^{3}}{( a ^{2} + x ^{2} ) ^{2}} d x

x \to - 1 - lim (\frac{x ^{2}}{1 + x ^{3}})

\int_{- 2}^{4} \frac{1}{2} d x