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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

\int x^{3} x^{2} - 81 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 = x^{2} - 81

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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