integral of x/(sqrt(8+2x-x^2))

Lösung

\int \frac{x}{8 + 2 x - x ^{2}} d x

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

Schritte zur Lösung

\int \frac{x}{8 + 2 x - x ^{2}} d x

Vervollständige das Quadrat

8 + 2 x - x^{2} : - (x - 1)^{2} + 9

= \int \frac{x}{- ( x - 1 ) ^{2} + 9} d x

Wende U-Substitution an

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

Multipliziere aus

\frac{u + 1}{- u ^{2} + 9} : \frac{u}{- u ^{2} + 9} + \frac{1}{- u ^{2} + 9}

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

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

\int \frac{1}{- u ^{2} + 9} d u = arcsin (\frac{1}{3} u)

= - - u^{2} + 9 + arcsin (\frac{1}{3} u)

Setze in

u = x - 1

ein

= - - (x - 1)^{2} + 9 + arcsin (\frac{1}{3} (x - 1))

Multipliziere aus

- (x - 1)^{2} + 9 : - x^{2} + 2 x + 8

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

Füge eine Konstante zur Lösung hinzu

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

\frac{\partial}{\partial y} (\frac{x z}{x ^{2} + y ^{2}})

\int 2 e^{7 x} d x

\int x^{3} + 1 d x

\int ln (1 - 2 x) d x

\int (x + 1)^{- \frac{1}{2}} d x