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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

x \to 3 lim (x^{2} - 2 x)

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

d er i v a t i v e y = \frac{t}{( t - 1 ) ^{2}}

x \to 8 lim (3 x)

t an g e n t f (x) = \frac{4}{5 x + 1}, at x = - 1