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Beliebt Rechnen >

integral of ysqrt(10+8y-16y^2)

Lösung

\int y 10 + 8 y - 16 y^{2} d y

Lösung

- \frac{1}{48} (- 16 y^{2} + 8 y + 10)^{\frac{3}{2}} + \frac{11}{32} (arcsin (\frac{1}{11} (4 y - 1)) + \frac{1}{2} sin (2 arcsin (\frac{4 y - 1}{11}))) + C

Schritte zur Lösung

\int y 10 + 8 y - 16 y^{2} d y

y 10 + 8 y - 16 y^{2} = 2 y 5 + 4 y - 8 y^{2}

= \int 2 y 5 + 4 y - 8 y^{2} d y

Entferne die Konstante:

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

= 2 \cdot \int y 5 + 4 y - 8 y^{2} d y

Vervollständige das Quadrat

5 + 4 y - 8 y^{2} : - 8 (y - \frac{1}{4})^{2} + \frac{11}{2}

= 2 \cdot \int y - 8 (y - \frac{1}{4})^{2} + \frac{11}{2} d y

Wende U-Substitution an

= 2 \cdot \int \frac{- 16 u ^{2} + 11}{2} (u + \frac{1}{4}) d u

\frac{- 16 u ^{2} + 11}{2} (u + \frac{1}{4}) = \frac{1}{2} - 16 u^{2} + 11 (u + \frac{1}{4})

= 2 \cdot \int \frac{1}{2} - 16 u^{2} + 11 (u + \frac{1}{4}) d u

Entferne die Konstante:

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

= 2 \frac{1}{2} \cdot \int - 16 u^{2} + 11 (u + \frac{1}{4}) d u

Multipliziere aus

- 16 u^{2} + 11 (u + \frac{1}{4}) : u - 16 u^{2} + 11 + \frac{- 16 u ^{2} + 11}{4}

= 2 \frac{1}{2} \cdot \int u - 16 u^{2} + 11 + \frac{- 16 u ^{2} + 11}{4} 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

= 2 \frac{1}{2} (\int u - 16 u^{2} + 11 d u + \int \frac{- 16 u ^{2} + 11}{4} d u)

\int u - 16 u^{2} + 11 d u = - \frac{1}{48} (- 16 u^{2} + 11)^{\frac{3}{2}}

\int \frac{- 16 u ^{2} + 11}{4} d u = \frac{11}{32} (arcsin (\frac{4}{11} u) + \frac{1}{2} sin (2 arcsin (\frac{4}{11} u)))

= 2 \frac{1}{2} (- \frac{1}{48} (- 16 u^{2} + 11)^{\frac{3}{2}} + \frac{11}{32} (arcsin (\frac{4}{11} u) + \frac{1}{2} sin (2 arcsin (\frac{4}{11} u))))

Setze in

u = y - \frac{1}{4}

ein

= 2 \frac{1}{2} - \frac{1}{48} (- 16 (y - \frac{1}{4})^{2} + 11)^{\frac{3}{2}} + \frac{11}{32} (arcsin (\frac{4}{11} (y - \frac{1}{4})) + \frac{1}{2} sin (2 arcsin (\frac{4}{11} (y - \frac{1}{4}))))

Vereinfache

2 \frac{1}{2} - \frac{1}{48} (- 16 (y - \frac{1}{4})^{2} + 11)^{\frac{3}{2}} + \frac{11}{32} (arcsin (\frac{4}{11} (y - \frac{1}{4})) + \frac{1}{2} sin (2 arcsin (\frac{4}{11} (y - \frac{1}{4})))) : - \frac{1}{48} (- 16 y^{2} + 8 y + 10)^{\frac{3}{2}} + \frac{11}{32} (arcsin (\frac{1}{11} (4 y - 1)) + \frac{1}{2} sin (2 arcsin (\frac{4 y - 1}{11})))

= - \frac{1}{48} (- 16 y^{2} + 8 y + 10)^{\frac{3}{2}} + \frac{11}{32} (arcsin (\frac{1}{11} (4 y - 1)) + \frac{1}{2} sin (2 arcsin (\frac{4 y - 1}{11})))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{48} (- 16 y^{2} + 8 y + 10)^{\frac{3}{2}} + \frac{11}{32} (arcsin (\frac{1}{11} (4 y - 1)) + \frac{1}{2} sin (2 arcsin (\frac{4 y - 1}{11}))) + C

Graph

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