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Beliebt Rechnen >

integral of x(1+sqrt(3+x^2))

Lösung

\int x (1 + 3 + x^{2}) d x

Lösung

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

Schritte zur Lösung

\int x (1 + 3 + x^{2}) d x

Wende U-Substitution an

= \int \frac{1 + u}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int 1 + u d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int 1 d u + \int u d u)

\int 1 d u = u

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

= \frac{1}{2} (u + \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = 3 + x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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