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Beliebt Rechnen >

integral of x^3sqrt(x^2-5)

Lösung

\int x^{3} x^{2} - 5 d x

Lösung

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

Schritte zur Lösung

\int x^{3} x^{2} - 5 d x

Wende U-Substitution an

= \int (u^{2} + 5) u^{2} d u

Multipliziere aus

(u^{2} + 5) u^{2} : u^{4} + 5 u^{2}

= \int u^{4} + 5 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 5 u^{2} d u

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

\int 5 u^{2} d u = \frac{5 u ^{3}}{3}

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

Setze in

u = x^{2} - 5

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

(dy)/(dx)-3/x y=x^3

\frac{d y}{d x} - \frac{3}{x} y = x^{3}

sum from k=1 to infinity of 0

k = 1 \sum \infty 0

derivative x^3+9x^2+35x+14

d er i v a t i v e x^{3} + 9 x^{2} + 35 x + 14

derivative (x^4)/4+1/(8x^2)

d er i v a t i v e \frac{x ^{4}}{4} + \frac{1}{8 x ^{2}}

derivative f(x)=(6x^2+4x+8)/(sqrt(x))

d er i v a t i v e f (x) = \frac{6 x ^{2} + 4 x + 8}{x}