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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

= \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

= \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}

= \frac{u ^{5}}{5} + 3 u^{3}

Setze in

u = x^{2} - 9

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

(dy)/(dx)=2^{x-y},y(-3)=-5

\frac{d y}{d x} = 2^{x - y}, y (- 3) = - 5

integral of (4x+8)/((x+5)^2)

\int \frac{4 x + 8}{( x + 5 ) ^{2}} d x

derivative 1/(4x^{3/4)}

d er i v a t i v e \frac{1}{4 x ^{\frac{3}{4}}}

steigung (-7.4)(-4.1)

s l o p e (- 7.4) (- 4.1)

t an g e n t - sin (x)