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

Lösung

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

- \frac{1}{x} + \frac{3}{2 x ^{2}} - \frac{1}{x ^{3}} + \frac{1}{4 x ^{4}} + C

Schritte zur Lösung

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

Multipliziere aus

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

= \int \frac{1}{x ^{2}} - \frac{3}{x ^{3}} + \frac{3}{x ^{4}} - \frac{1}{x ^{5}} 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

= \int \frac{1}{x ^{2}} d x - \int \frac{3}{x ^{3}} d x + \int \frac{3}{x ^{4}} d x - \int \frac{1}{x ^{5}} d x

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

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

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

\int \frac{1}{x ^{5}} d x = - \frac{1}{4 x ^{4}}

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

Vereinfache

= - \frac{1}{x} + \frac{3}{2 x ^{2}} - \frac{1}{x ^{3}} + \frac{1}{4 x ^{4}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{x} + \frac{3}{2 x ^{2}} - \frac{1}{x ^{3}} + \frac{1}{4 x ^{4}} + C

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

\int (\frac{10}{y ^{\frac{1}{4}}} + y) d y

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

\int \frac{5 ln ( x )}{x 4 - ln ^{4} ( x )} d x

\int \frac{( ( ln ( x ) ) ^{42} )}{x} d x