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Beliebt Rechnen >

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

Lösung

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

Lösung

\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

Entferne die Konstante:

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

= - \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

Graph

Beliebte Beispiele

integral of (cos(3x))/(sin(3x))

\int \frac{cos ( 3 x )}{sin ( 3 x )} d x

integral of 1/((u^2+a^2)^2)

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

y^{''}+10y^'+26y=0

y^{^{''}} + 10 y^{^{'}} + 26 y = 0

derivative f(x)=x^2sqrt(9-x^2)

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x^4+y^3sqrt(x^5+3)y^'=0

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