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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vervollständige das Quadrat

2 + x - x^{2} : - (x - \frac{1}{2})^{2} + \frac{9}{4}

= \int \frac{x}{- ( x - \frac{1}{2} ) ^{2} + \frac{9}{4}} d x

Wende U-Substitution an

= \int \frac{2 u + 1}{- 4 u ^{2} + 9} d u

Multipliziere aus

\frac{2 u + 1}{- 4 u ^{2} + 9} : \frac{2 u}{- 4 u ^{2} + 9} + \frac{1}{- 4 u ^{2} + 9}

= \int \frac{2 u}{- 4 u ^{2} + 9} + \frac{1}{- 4 u ^{2} + 9} 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{2 u}{- 4 u ^{2} + 9} d u + \int \frac{1}{- 4 u ^{2} + 9} d u

\int \frac{2 u}{- 4 u ^{2} + 9} d u = - \frac{1}{2} - 4 u^{2} + 9

\int \frac{1}{- 4 u ^{2} + 9} d u = \frac{1}{2} arcsin (\frac{2}{3} u)

= - \frac{1}{2} - 4 u^{2} + 9 + \frac{1}{2} arcsin (\frac{2}{3} u)

Setze in

u = x - \frac{1}{2}

ein

= - \frac{1}{2} - 4 (x - \frac{1}{2})^{2} + 9 + \frac{1}{2} arcsin (\frac{2}{3} (x - \frac{1}{2}))

Vereinfache

- \frac{1}{2} - 4 (x - \frac{1}{2})^{2} + 9 + \frac{1}{2} arcsin (\frac{2}{3} (x - \frac{1}{2})) : - - x^{2} + x + 2 + \frac{1}{2} arcsin (\frac{2}{3} x - \frac{1}{3})

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int \frac{t ^{2}}{( 1 + t ^{2} ) ^{2}} d t

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