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integral of sqrt(12+8x-3x^2)

Lösung

\int 12 + 8 x - 3 x^{2} d x

\frac{13}{3 3} (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}) + sin (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}))) + C

Schritte zur Lösung

\int 12 + 8 x - 3 x^{2} d x

Vervollständige das Quadrat

12 + 8 x - 3 x^{2} : - 3 (x - \frac{4}{3})^{2} + \frac{52}{3}

= \int - 3 (x - \frac{4}{3})^{2} + \frac{52}{3} d x

Wende U-Substitution an

= \int \frac{- 9 u ^{2} + 52}{3} d u

\frac{- 9 u ^{2} + 52}{3} = \frac{1}{3} - 9 u^{2} + 52

= \int \frac{1}{3} - 9 u^{2} + 52 d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int - 9 u^{2} + 52 d u

Trigonometrische Substitution anwenden

= \frac{1}{3} \cdot \int \frac{52}{3} cos^{2} (v) d v

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \frac{52}{3} \cdot \int cos^{2} (v) d v

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{1}{3} \frac{52}{3} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= \frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Ersetze zurück

= \frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} (arcsin (\frac{3}{2 13} (x - \frac{4}{3})) + \frac{1}{2} sin (2 arcsin (\frac{3}{2 13} (x - \frac{4}{3}))))

Vereinfache

\frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} (arcsin (\frac{3}{2 13} (x - \frac{4}{3})) + \frac{1}{2} sin (2 arcsin (\frac{3}{2 13} (x - \frac{4}{3})))) : \frac{13}{3 3} (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}) + sin (2 arcsin (\frac{3}{2 13} x - \frac{2}{13})))

= \frac{13}{3 3} (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}) + sin (2 arcsin (\frac{3}{2 13} x - \frac{2}{13})))

Füge eine Konstante zur Lösung hinzu

= \frac{13}{3 3} (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}) + sin (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}))) + C

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