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Beliebt Rechnen >

integral of (x^2)/(sqrt(10x-x^2))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vervollständige das Quadrat

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

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

Wende U-Substitution an

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

Multipliziere aus

\frac{( u + 5 ) ^{2}}{- u ^{2} + 25} : \frac{u ^{2}}{- u ^{2} + 25} + \frac{10 u}{- u ^{2} + 25} + \frac{25}{- u ^{2} + 25}

= \int \frac{u ^{2}}{- u ^{2} + 25} + \frac{10 u}{- u ^{2} + 25} + \frac{25}{- 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 \frac{u ^{2}}{- u ^{2} + 25} d u + \int \frac{10 u}{- u ^{2} + 25} d u + \int \frac{25}{- u ^{2} + 25} d u

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

\int \frac{10 u}{- u ^{2} + 25} d u = - 10 - u^{2} + 25

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

= \frac{25}{2} (arcsin (\frac{1}{5} u) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} u))) - 10 - u^{2} + 25 + 25 arcsin (\frac{1}{5} u)

Setze in

u = x - 5

ein

= \frac{25}{2} (arcsin (\frac{1}{5} (x - 5)) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} (x - 5)))) - 10 - (x - 5)^{2} + 25 + 25 arcsin (\frac{1}{5} (x - 5))

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= \frac{25}{4} (2 arcsin (\frac{1}{5} x - 1) - sin (2 arcsin (\frac{1}{5} x - 1))) - 10 - x^{2} + 10 x + 25 arcsin (\frac{1}{5} x - 1) + C

Graph

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