integral of x/(sqrt(-46-6x^2+36x))

Lösung

\int \frac{x}{- 46 - 6 x ^{2} + 36 x} d x

- \frac{1}{6} - 6 x^{2} + 36 x - 46 + \frac{3}{2} arcsin (\frac{3}{2} (x - 3)) + C

Schritte zur Lösung

\int \frac{x}{- 46 - 6 x ^{2} + 36 x} d x

Vervollständige das Quadrat

- 46 - 6 x^{2} + 36 x : - 6 (x - 3)^{2} + 8

= \int \frac{x}{- 6 ( x - 3 ) ^{2} + 8} d x

Wende U-Substitution an

= \int \frac{u + 3}{- 6 u ^{2} + 8} d u

Multipliziere aus

\frac{u + 3}{- 6 u ^{2} + 8} : \frac{u}{- 6 u ^{2} + 8} + \frac{3}{- 6 u ^{2} + 8}

= \int \frac{u}{- 6 u ^{2} + 8} + \frac{3}{- 6 u ^{2} + 8} 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}{- 6 u ^{2} + 8} d u + \int \frac{3}{- 6 u ^{2} + 8} d u

\int \frac{u}{- 6 u ^{2} + 8} d u = - \frac{1}{6} - 6 u^{2} + 8

\int \frac{3}{- 6 u ^{2} + 8} d u = \frac{3}{2} arcsin (\frac{3}{2} u)

= - \frac{1}{6} - 6 u^{2} + 8 + \frac{3}{2} arcsin (\frac{3}{2} u)

Setze in

u = x - 3

ein

= - \frac{1}{6} - 6 (x - 3)^{2} + 8 + \frac{3}{2} arcsin (\frac{3}{2} (x - 3))

Vereinfache

- \frac{1}{6} - 6 (x - 3)^{2} + 8 + \frac{3}{2} arcsin (\frac{3}{2} (x - 3)) : - \frac{1}{6} - 6 x^{2} + 36 x - 46 + \frac{3}{2} arcsin (\frac{3}{2} (x - 3))

= - \frac{1}{6} - 6 x^{2} + 36 x - 46 + \frac{3}{2} arcsin (\frac{3}{2} (x - 3))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{6} - 6 x^{2} + 36 x - 46 + \frac{3}{2} arcsin (\frac{3}{2} (x - 3)) + C

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