integral of (-2x^4)/((1-x)^3)

Lösung

\int \frac{- 2 x ^{4}}{( 1 - x ) ^{3}} d x

- \frac{x ^{4}}{( 1 - x ) ^{2}} + \frac{4 x ^{3}}{1 - x} + 6 x^{2} + 12 x + 12 ln ∣ - x + 1 ∣ - 18 + C

Schritte zur Lösung

\int \frac{- 2 x ^{4}}{( 1 - x ) ^{3}} 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}}{( 1 - x ) ^{3}} d x

Wende die partielle Integration an

= - 2 (\frac{x ^{4}}{2 ( 1 - x ) ^{2}} - \int \frac{2 x ^{3}}{( 1 - x ) ^{2}} d x)

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

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

Vereinfache

- 2 (\frac{x ^{4}}{2 ( 1 - x ) ^{2}} - 2 (\frac{x ^{3}}{1 - x} + 3 (\frac{1}{2} x^{2} + x + ln ∣ - x + 1 ∣ + \frac{1}{2} - 2))) : - \frac{x ^{4}}{( 1 - x ) ^{2}} + \frac{4 x ^{3}}{1 - x} + 6 x^{2} + 12 x + 12 ln ∣ - x + 1 ∣ - 18

= - \frac{x ^{4}}{( 1 - x ) ^{2}} + \frac{4 x ^{3}}{1 - x} + 6 x^{2} + 12 x + 12 ln ∣ - x + 1 ∣ - 18

Füge eine Konstante zur Lösung hinzu

= - \frac{x ^{4}}{( 1 - x ) ^{2}} + \frac{4 x ^{3}}{1 - x} + 6 x^{2} + 12 x + 12 ln ∣ - x + 1 ∣ - 18 + C

ma c l a u r in sin (x^{3})

f (t) = 2 t

x \to 1 lim (\frac{x - 1}{x ^{6} - 1})

y^{^{''}} + 4 y = t^{2}

d er i v a t i v e f (x) = e^{\infty}