integral of 4x(1+sqrt(x))

Lösung

\int 4 x (1 + x) d x

4 (\frac{x ^{2}}{2} + \frac{2}{5} x^{\frac{5}{2}}) + C

Schritte zur Lösung

\int 4 x (1 + x) d x

Entferne die Konstante:

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

= 4 \cdot \int x (1 + x) d x

Multipliziere aus

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

= 4 \cdot \int x + x^{\frac{3}{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

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

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

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

= 4 (\frac{x ^{2}}{2} + \frac{2}{5} x^{\frac{5}{2}})

Füge eine Konstante zur Lösung hinzu

= 4 (\frac{x ^{2}}{2} + \frac{2}{5} x^{\frac{5}{2}}) + C

n = 0 \sum \infty \frac{1}{10}

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