integral of sqrt(x)*(3x+sqrt(x))

Lösung

\int x \cdot (3 x + x) d x

\frac{6}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2} + C

Schritte zur Lösung

\int x (3 x + x) d x

Multipliziere aus

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

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

\int 3 x^{\frac{3}{2}} d x = \frac{6}{5} x^{\frac{5}{2}}

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

= \frac{6}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{6}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2} + C

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