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Beliebt Rechnen >

integral of x^3sqrt(64-x^2)

Lösung

\int x^{3} 64 - x^{2} d x

Lösung

\frac{1}{5} (64 - x^{2})^{\frac{5}{2}} - \frac{64}{3} (64 - x^{2})^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{3} 64 - x^{2} d x

Wende U-Substitution an

= \int - (- u^{2} + 64) u^{2} d u

Entferne die Konstante:

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

= - \int (- u^{2} + 64) u^{2} d u

Multipliziere aus

(- u^{2} + 64) u^{2} : - u^{4} + 64 u^{2}

= - \int - u^{4} + 64 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 64 u^{2} d u)

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

\int 64 u^{2} d u = \frac{64 u ^{3}}{3}

= - (- \frac{u ^{5}}{5} + \frac{64 u ^{3}}{3})

Setze in

u = 64 - x^{2}

ein

= - (- \frac{( 64 - x ^{2} ) ^{5}}{5} + \frac{64 ( 64 - x ^{2} ) ^{3}}{3})

Vereinfache

- (- \frac{( 64 - x ^{2} ) ^{5}}{5} + \frac{64 ( 64 - x ^{2} ) ^{3}}{3}) : \frac{1}{5} (64 - x^{2})^{\frac{5}{2}} - \frac{64}{3} (64 - x^{2})^{\frac{3}{2}}

= \frac{1}{5} (64 - x^{2})^{\frac{5}{2}} - \frac{64}{3} (64 - x^{2})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (64 - x^{2})^{\frac{5}{2}} - \frac{64}{3} (64 - x^{2})^{\frac{3}{2}} + C

Graph

Beliebte Beispiele

integral of xsqrt(81+x^2)

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

fläche y=sqrt(2x),y=12-x,y=4-x

a re a y = 2 x, y = 12 - x, y = 4 - x

tangent f(x)=(2x-1)/(x+9),\at x=2

t an g e n t f (x) = \frac{2 x - 1}{x + 9}, at x = 2

derivative y=arccsc(4x)

d er i v a t i v e y = arccsc (4 x)

xy^{''}+2y^'=8x^2

x y^{^{''}} + 2 y^{^{'}} = 8 x^{2}