integral of (u^2)/(sqrt(u^2-8u))

Lösung

\int \frac{u ^{2}}{u ^{2} - 8 u} d u

\frac{1}{2} (u - 4) u^{2} - 8 u + 24 ln (\frac{1}{4} u + u^{2} - 8 u - 4) + 8 u^{2} - 8 u + C

Schritte zur Lösung

\int \frac{u ^{2}}{u ^{2} - 8 u} d u

Vervollständige das Quadrat

u^{2} - 8 u : (u - 4)^{2} - 16

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

Wende U-Substitution an

= \int \frac{( v + 4 ) ^{2}}{v ^{2} - 16} d v

Multipliziere aus

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

= \int \frac{v ^{2}}{v ^{2} - 16} + \frac{8 v}{v ^{2} - 16} + \frac{16}{v ^{2} - 16} 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

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

\int \frac{v ^{2}}{v ^{2} - 16} d v = 16 (\frac{1}{32} v v^{2} - 16 + \frac{1}{2} ln (\frac{v ^{2} - 16 + v}{4}))

\int \frac{8 v}{v ^{2} - 16} d v = 8 v^{2} - 16

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

= 16 (\frac{1}{32} v v^{2} - 16 + \frac{1}{2} ln (\frac{v ^{2} - 16 + v}{4})) + 8 v^{2} - 16 + 16 ln (\frac{1}{4} v^{2} - 16 + v)

Setze in

v = u - 4

ein

= 16 \frac{1}{32} (u - 4) (u - 4)^{2} - 16 + \frac{1}{2} ln \frac{( u - 4 ) ^{2} - 16 + u - 4}{4} + 8 (u - 4)^{2} - 16 + 16 ln (\frac{1}{4} (u - 4)^{2} - 16 + u - 4)

Vereinfache

16 \frac{1}{32} (u - 4) (u - 4)^{2} - 16 + \frac{1}{2} ln \frac{( u - 4 ) ^{2} - 16 + u - 4}{4} + 8 (u - 4)^{2} - 16 + 16 ln (\frac{1}{4} (u - 4)^{2} - 16 + u - 4) : \frac{1}{2} (u - 4) u^{2} - 8 u + 24 ln (\frac{1}{4} u + u^{2} - 8 u - 4) + 8 u^{2} - 8 u

= \frac{1}{2} (u - 4) u^{2} - 8 u + 24 ln (\frac{1}{4} u + u^{2} - 8 u - 4) + 8 u^{2} - 8 u

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (u - 4) u^{2} - 8 u + 24 ln (\frac{1}{4} u + u^{2} - 8 u - 4) + 8 u^{2} - 8 u + C

\frac{d}{d x} (x^{\frac{1}{3}} (x + 4))

d er i v a t i v e \frac{2}{1 + 3 x + x ^{2}}

(24 + x^{2}) y^{^{'}} - 2 x y = 0

\frac{d y}{d x} = \frac{y ^{3}}{x}, y (1) = 1

\int \frac{x ^{2} - 5 x + 9}{x ^{2} - 5 x + 6} d x