integral of sqrt(1-2x)(x+2)

Lösung

\int 1 - 2 x (x + 2) d x

- \frac{1}{5} x (- 2 x + 1)^{\frac{3}{2}} - \frac{11}{15} (1 - 2 x)^{\frac{3}{2}} + C

Schritte zur Lösung

\int 1 - 2 x (x + 2) d x

Multipliziere aus

1 - 2 x (x + 2) : x 1 - 2 x + 2 1 - 2 x

= \int x 1 - 2 x + 2 1 - 2 x 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 1 - 2 x d x + \int 2 1 - 2 x d x

\int x 1 - 2 x d x = - \frac{1}{15} (3 x + 1) (- 2 x + 1)^{\frac{3}{2}}

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

= - \frac{1}{15} (3 x + 1) (- 2 x + 1)^{\frac{3}{2}} - \frac{2}{3} (1 - 2 x)^{\frac{3}{2}}

Vereinfache

- \frac{1}{15} (3 x + 1) (- 2 x + 1)^{\frac{3}{2}} - \frac{2}{3} (1 - 2 x)^{\frac{3}{2}} : - \frac{1}{5} x (- 2 x + 1)^{\frac{3}{2}} - \frac{11}{15} (1 - 2 x)^{\frac{3}{2}}

= - \frac{1}{5} x (- 2 x + 1)^{\frac{3}{2}} - \frac{11}{15} (1 - 2 x)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{5} x (- 2 x + 1)^{\frac{3}{2}} - \frac{11}{15} (1 - 2 x)^{\frac{3}{2}} + C

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