integral of (x^8+x+1+x^{-1}+x^{-2})

Lösung

\int (x^{8} + x + 1 + x^{- 1} + x^{- 2}) d x

\frac{x ^{9}}{9} + \frac{x ^{2}}{2} + x + ln ∣ x ∣ - \frac{1}{x} + C

Schritte zur Lösung

\int x^{8} + x + 1 + x^{- 1} + x^{- 2} 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 x^{8} d x + \int x d x + \int 1 d x + \int x^{- 1} d x + \int x^{- 2} d x

\int x^{8} d x = \frac{x ^{9}}{9}

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

\int 1 d x = x

\int x^{- 1} d x = ln ∣ x ∣

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

= \frac{x ^{9}}{9} + \frac{x ^{2}}{2} + x + ln ∣ x ∣ - \frac{1}{x}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{9}}{9} + \frac{x ^{2}}{2} + x + ln ∣ x ∣ - \frac{1}{x} + C

\int (21 x^{2} - 12 x - 5) d x

\int (x^{4} + 3) (3 + 2 x) d x

\int y^{10} d y

\int 8 - 2 x^{2} d x

\int \frac{e ^{x}}{( e ^{x} + 1 ) ^{6}} d x