integral of (x+sqrt(x))^2

Lösung

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

\frac{x ^{3}}{3} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2} + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

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

= \frac{x ^{3}}{3} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2} + C

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

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