integral of (9w^8-5w^5+3w^2-16)/(8w^4)

Lösung

\int \frac{9 w ^{8} - 5 w ^{5} + 3 w ^{2} - 16}{8 w ^{4}} d w

\frac{1}{8} (\frac{9 w ^{5}}{5} - \frac{5 w ^{2}}{2} - \frac{3}{w} + \frac{16}{3 w ^{3}}) + C

Schritte zur Lösung

\int \frac{9 w ^{8} - 5 w ^{5} + 3 w ^{2} - 16}{8 w ^{4}} d w

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int \frac{9 w ^{8} - 5 w ^{5} + 3 w ^{2} - 16}{w ^{4}} d w

Multipliziere aus

\frac{9 w ^{8} - 5 w ^{5} + 3 w ^{2} - 16}{w ^{4}} : 9 w^{4} - 5 w + \frac{3}{w ^{2}} - \frac{16}{w ^{4}}

= \frac{1}{8} \cdot \int 9 w^{4} - 5 w + \frac{3}{w ^{2}} - \frac{16}{w ^{4}} d w

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{8} (\int 9 w^{4} d w - \int 5 w d w + \int \frac{3}{w ^{2}} d w - \int \frac{16}{w ^{4}} d w)

\int 9 w^{4} d w = \frac{9 w ^{5}}{5}

\int 5 w d w = \frac{5 w ^{2}}{2}

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

\int \frac{16}{w ^{4}} d w = - \frac{16}{3 w ^{3}}

= \frac{1}{8} (\frac{9 w ^{5}}{5} - \frac{5 w ^{2}}{2} - \frac{3}{w} - (- \frac{16}{3 w ^{3}}))

Vereinfache

= \frac{1}{8} (\frac{9 w ^{5}}{5} - \frac{5 w ^{2}}{2} - \frac{3}{w} + \frac{16}{3 w ^{3}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{8} (\frac{9 w ^{5}}{5} - \frac{5 w ^{2}}{2} - \frac{3}{w} + \frac{16}{3 w ^{3}}) + C

\int - 5 x^{- 8} d x

\frac{d}{d x} (- \frac{x ^{3} + 8 x ^{2} + 19 x + 12}{( x + 20 )})

\frac{d}{d x} (a \cdot x + b)

\int \frac{4 x - 5}{4 x ^{2} - 12 x + 13} d x

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