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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vervollständige das Quadrat

2 x - x^{2} : - (x - 1)^{2} + 1

= \int \frac{x ^{2}}{- ( x - 1 ) ^{2} + 1} d x

Wende U-Substitution an

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

Multipliziere aus

\frac{( u + 1 ) ^{2}}{- u ^{2} + 1} : \frac{u ^{2}}{- u ^{2} + 1} + \frac{2 u}{- u ^{2} + 1} + \frac{1}{- u ^{2} + 1}

= \int \frac{u ^{2}}{- u ^{2} + 1} + \frac{2 u}{- u ^{2} + 1} + \frac{1}{- u ^{2} + 1} 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} + 1} d u + \int \frac{2 u}{- u ^{2} + 1} d u + \int \frac{1}{- u ^{2} + 1} d u

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

\int \frac{2 u}{- u ^{2} + 1} d u = - 2 - u^{2} + 1

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

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

Setze in

u = x - 1

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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