integral of (-2x^4)/(2x^3-3*x^2)

Lösung

\int \frac{- 2 \cdot x ^{4}}{2 \cdot x ^{3} - 3 \cdot x ^{2}} d x

- 2 (\frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣) + C

Schritte zur Lösung

\int \frac{- 2 x ^{4}}{2 x ^{3} - 3 x ^{2}} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int \frac{x ^{4}}{2 x ^{3} - 3 x ^{2}} d x

Vereinfache

\frac{x ^{4}}{2 x ^{3} - 3 x ^{2}} : \frac{x ^{2}}{2 x - 3}

= - 2 \cdot \int \frac{x ^{2}}{2 x - 3} d x

Lange Division

\frac{x ^{2}}{2 x - 3} : \frac{x}{2} + \frac{3}{4} + \frac{9}{4 ( 2 x - 3 )}

= - 2 \cdot \int \frac{x}{2} + \frac{3}{4} + \frac{9}{4 ( 2 x - 3 )} 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

= - 2 (\int \frac{x}{2} d x + \int \frac{3}{4} d x + \int \frac{9}{4 ( 2 x - 3 )} d x)

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

\int \frac{3}{4} d x = \frac{3}{4} x

\int \frac{9}{4 ( 2 x - 3 )} d x = \frac{9}{8} ln ∣ 2 x - 3 ∣

= - 2 (\frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣)

Füge eine Konstante zur Lösung hinzu

= - 2 (\frac{x ^{2}}{4} + \frac{3}{4} x + \frac{9}{8} ln ∣ 2 x - 3 ∣) + C

\int x^{2} (1 - x)^{\frac{1}{2}} d x

\frac{d y}{d x} + 6 y = 0

\int_{2}^{3} x^{5} d x

a re a f (x) = x^{2} - 2 x + 1, g (x) = - x + 3

\int csc^{3} (x) cot (x) d x