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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{( u - 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 (u - 1) u d u

Multipliziere aus

(u - 1) u : u^{\frac{3}{2}} - u

= \frac{1}{2} \cdot \int u^{\frac{3}{2}} - 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 u^{\frac{3}{2}} d u - \int u d u)

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

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

= \frac{1}{2} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Setze in

u = 1 + x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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