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Beliebt Rechnen >

integral of (y^2)/(sqrt(9-8y-y^2))

Lösung

\int \frac{y ^{2}}{9 - 8 y - y ^{2}} d y

Lösung

\frac{57}{2} arcsin (\frac{1}{5} (y + 4)) - \frac{25}{4} sin (2 arcsin (\frac{1}{5} (y + 4))) + 8 - y^{2} - 8 y + 9 + C

Schritte zur Lösung

\int \frac{y ^{2}}{9 - 8 y - y ^{2}} d y

Vervollständige das Quadrat

9 - 8 y - y^{2} : - (y + 4)^{2} + 25

= \int \frac{y ^{2}}{- ( y + 4 ) ^{2} + 25} d y

Wende U-Substitution an

= \int \frac{( u - 4 ) ^{2}}{- u ^{2} + 25} d u

Multipliziere aus

\frac{( u - 4 ) ^{2}}{- u ^{2} + 25} : \frac{u ^{2}}{- u ^{2} + 25} - \frac{8 u}{- u ^{2} + 25} + \frac{16}{- u ^{2} + 25}

= \int \frac{u ^{2}}{- u ^{2} + 25} - \frac{8 u}{- u ^{2} + 25} + \frac{16}{- u ^{2} + 25} 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} + 25} d u - \int \frac{8 u}{- u ^{2} + 25} d u + \int \frac{16}{- u ^{2} + 25} d u

\int \frac{u ^{2}}{- u ^{2} + 25} d u = \frac{25}{2} (arcsin (\frac{1}{5} u) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} u)))

\int \frac{8 u}{- u ^{2} + 25} d u = - 8 - u^{2} + 25

\int \frac{16}{- u ^{2} + 25} d u = 16 arcsin (\frac{1}{5} u)

= \frac{25}{2} (arcsin (\frac{1}{5} u) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} u))) - (- 8 - u^{2} + 25) + 16 arcsin (\frac{1}{5} u)

Setze in

u = y + 4

ein

= \frac{25}{2} (arcsin (\frac{1}{5} (y + 4)) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} (y + 4)))) - (- 8 - (y + 4)^{2} + 25) + 16 arcsin (\frac{1}{5} (y + 4))

Vereinfache

\frac{25}{2} (arcsin (\frac{1}{5} (y + 4)) - \frac{1}{2} sin (2 arcsin (\frac{1}{5} (y + 4)))) - (- 8 - (y + 4)^{2} + 25) + 16 arcsin (\frac{1}{5} (y + 4)) : \frac{57}{2} arcsin (\frac{1}{5} (y + 4)) - \frac{25}{4} sin (2 arcsin (\frac{1}{5} (y + 4))) + 8 - y^{2} - 8 y + 9

= \frac{57}{2} arcsin (\frac{1}{5} (y + 4)) - \frac{25}{4} sin (2 arcsin (\frac{1}{5} (y + 4))) + 8 - y^{2} - 8 y + 9

Füge eine Konstante zur Lösung hinzu

= \frac{57}{2} arcsin (\frac{1}{5} (y + 4)) - \frac{25}{4} sin (2 arcsin (\frac{1}{5} (y + 4))) + 8 - y^{2} - 8 y + 9 + C

Graph

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