integral of 1/(x(1-sqrt(x))^2)

Lösung

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

2 (\frac{1}{( - x + 1 ) ^{2}} - \frac{1}{( x - 1 ) ^{2}} + ln x - ln x - 1 - \frac{1}{x - 1}) + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{2}{u ( - u + 1 ) ^{2}} d u

Entferne die Konstante:

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

= 2 \cdot \int \frac{1}{u ( - u + 1 ) ^{2}} d u

Wende die partielle Integration an

= 2 (\frac{1}{( - u + 1 ) ^{2}} - \int - \frac{3 u - 1}{u ( u - 1 ) ^{3}} d u)

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

= 2 (\frac{1}{( - u + 1 ) ^{2}} - (\frac{1}{( u - 1 ) ^{2}} - ln ∣ u ∣ + ln ∣ u - 1 ∣ + \frac{1}{u - 1}))

Setze in

u = x

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{( - x + 1 ) ^{2}} - \frac{1}{( x - 1 ) ^{2}} + ln x - ln x - 1 - \frac{1}{x - 1}) + C

\int_{0}^{4} x d x

\frac{d}{d x} (\frac{132}{x ^{13}})

\int_{0}^{\infty} 3 x e^{- 3 x} d x

x \to 0 lim (\frac{csc ( x )}{3 x})

\frac{\partial}{\partial z} (x^{2} y + y z^{3} + z^{5} x)