integral of-8x^3sqrt(9-x^2)

Lösung

\int - 8 x^{3} 9 - x^{2} d x

8 (- \frac{1}{5} (9 - x^{2})^{\frac{5}{2}} + 3 (9 - x^{2})^{\frac{3}{2}}) + C

Schritte zur Lösung

\int - 8 x^{3} 9 - x^{2} d x

Entferne die Konstante:

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

= - 8 \cdot \int x^{3} 9 - x^{2} d x

Wende U-Substitution an

= - 8 \cdot \int - (- u^{2} + 9) u^{2} d u

Entferne die Konstante:

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

= - 8 (- \int (- u^{2} + 9) u^{2} d u)

Multipliziere aus

(- u^{2} + 9) u^{2} : - u^{4} + 9 u^{2}

= - 8 (- \int - u^{4} + 9 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

= - 8 (- (- \int u^{4} d u + \int 9 u^{2} d u))

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

\int 9 u^{2} d u = 3 u^{3}

= - 8 (- (- \frac{u ^{5}}{5} + 3 u^{3}))

Setze in

u = 9 - x^{2}

ein

= - 8 (- (- \frac{( 9 - x ^{2} ) ^{5}}{5} + 3 (9 - x^{2})^{3}))

Vereinfache

- 8 (- (- \frac{( 9 - x ^{2} ) ^{5}}{5} + 3 (9 - x^{2})^{3})) : 8 (- \frac{1}{5} (9 - x^{2})^{\frac{5}{2}} + 3 (9 - x^{2})^{\frac{3}{2}})

= 8 (- \frac{1}{5} (9 - x^{2})^{\frac{5}{2}} + 3 (9 - x^{2})^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= 8 (- \frac{1}{5} (9 - x^{2})^{\frac{5}{2}} + 3 (9 - x^{2})^{\frac{3}{2}}) + C

\int x (x^{2} - 8) d x

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

\int x^{2} - 50 x + 700 d x

\int (7 x^{5} + 1)^{9} x^{4} d x

\int (4 x^{3} + \frac{81}{8 x ^{4}}) d x