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

Lösung

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

- \frac{2 x - 1}{3 x ^{\frac{3}{2}}} + \frac{2}{3} \frac{x - 1}{x} + C

Schritte zur Lösung

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

\frac{1}{x ^{2}} = \frac{1 - x ^{2}}{x ^{2}} + 1

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

Multipliziere aus

(\frac{1 - x ^{2}}{x ^{2}} + 1) 1 - \frac{1}{x} : \frac{x - 1}{x ^{5}}

= \int \frac{x - 1}{x ^{5}} d x

Vereinfache

\frac{x - 1}{x ^{5}} : \frac{x - 1}{x ^{5}}

= \int \frac{x - 1}{x ^{5}} d x

Wende die partielle Integration an

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

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

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

Vereinfache

= - \frac{2 x - 1}{3 x ^{\frac{3}{2}}} + \frac{2}{3} \frac{x - 1}{x}

Füge eine Konstante zur Lösung hinzu

= - \frac{2 x - 1}{3 x ^{\frac{3}{2}}} + \frac{2}{3} \frac{x - 1}{x} + C

\frac{d y}{d x} = y cos (3 x + 2)

\frac{d y}{d t} + 7 y = e^{5 t}

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

x^{^{''}} + 3 x^{^{'}} + 2 x = 1

y^{^{'}} = \frac{1}{x} y - \frac{x ^{2}}{3} y^{4}, y (1) = 1