integral of xsqrt(10x-x^2)

Lösung

\int x 10 x - x^{2} d x

- \frac{1}{3} (- x^{2} + 10 x)^{\frac{3}{2}} + \frac{125}{4} (2 arcsin (\frac{1}{5} x - 1) + sin (2 arcsin (\frac{1}{5} x - 1))) + C

Schritte zur Lösung

\int x 10 x - x^{2} d x

Vervollständige das Quadrat

10 x - x^{2} : - (x - 5)^{2} + 25

= \int x - (x - 5)^{2} + 25 d x

Wende U-Substitution an

= \int (u + 5) - u^{2} + 25 d u

Multipliziere aus

(u + 5) - u^{2} + 25 : u - u^{2} + 25 + 5 - u^{2} + 25

= \int u - u^{2} + 25 + 5 - u^{2} + 25 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 - u^{2} + 25 d u + \int 5 - u^{2} + 25 d u

\int u - u^{2} + 25 d u = - \frac{1}{3} (- u^{2} + 25)^{\frac{3}{2}}

\int 5 - u^{2} + 25 d u = \frac{125}{2} (arcsin (\frac{1}{5} u) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} u)))

= - \frac{1}{3} (- u^{2} + 25)^{\frac{3}{2}} + \frac{125}{2} (arcsin (\frac{1}{5} u) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} u)))

Setze in

u = x - 5

ein

= - \frac{1}{3} (- (x - 5)^{2} + 25)^{\frac{3}{2}} + \frac{125}{2} (arcsin (\frac{1}{5} (x - 5)) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} (x - 5))))

Vereinfache

- \frac{1}{3} (- (x - 5)^{2} + 25)^{\frac{3}{2}} + \frac{125}{2} (arcsin (\frac{1}{5} (x - 5)) + \frac{1}{2} sin (2 arcsin (\frac{1}{5} (x - 5)))) : - \frac{1}{3} (- x^{2} + 10 x)^{\frac{3}{2}} + \frac{125}{4} (2 arcsin (\frac{1}{5} x - 1) + sin (2 arcsin (\frac{1}{5} x - 1)))

= - \frac{1}{3} (- x^{2} + 10 x)^{\frac{3}{2}} + \frac{125}{4} (2 arcsin (\frac{1}{5} x - 1) + sin (2 arcsin (\frac{1}{5} x - 1)))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{3} (- x^{2} + 10 x)^{\frac{3}{2}} + \frac{125}{4} (2 arcsin (\frac{1}{5} x - 1) + sin (2 arcsin (\frac{1}{5} x - 1))) + C

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