integral of x^3sqrt(81+x^2)

Lösung

\int x^{3} 81 + x^{2} d x

\frac{1}{5} (81 + x^{2})^{\frac{5}{2}} - 27 (81 + x^{2})^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{3} 81 + x^{2} d x

Wende U-Substitution an

= \int (u^{2} - 81) u^{2} d u

Multipliziere aus

(u^{2} - 81) u^{2} : u^{4} - 81 u^{2}

= \int u^{4} - 81 u^{2} 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

= \int u^{4} d u - \int 81 u^{2} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int 81 u^{2} d u = 27 u^{3}

= \frac{u ^{5}}{5} - 27 u^{3}

Setze in

u = 81 + x^{2}

ein

= \frac{( 81 + x ^{2} ) ^{5}}{5} - 27 (81 + x^{2})^{3}

Vereinfache

\frac{( 81 + x ^{2} ) ^{5}}{5} - 27 (81 + x^{2})^{3} : \frac{1}{5} (81 + x^{2})^{\frac{5}{2}} - 27 (81 + x^{2})^{\frac{3}{2}}

= \frac{1}{5} (81 + x^{2})^{\frac{5}{2}} - 27 (81 + x^{2})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (81 + x^{2})^{\frac{5}{2}} - 27 (81 + x^{2})^{\frac{3}{2}} + C

\frac{\partial}{\partial x} (e^{- \frac{x}{2 y}})

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