integral of x/(sqrt(-x^2-8x-7))

Lösung

\int \frac{x}{- x ^{2} - 8 x - 7} d x

- - x^{2} - 8 x - 7 - 4 arcsin (\frac{1}{3} (x + 4)) + C

Schritte zur Lösung

\int \frac{x}{- x ^{2} - 8 x - 7} d x

Vervollständige das Quadrat

- x^{2} - 8 x - 7 : - (x + 4)^{2} + 9

= \int \frac{x}{- ( x + 4 ) ^{2} + 9} d x

Wende U-Substitution an

= \int \frac{u - 4}{- u ^{2} + 9} d u

Multipliziere aus

\frac{u - 4}{- u ^{2} + 9} : \frac{u}{- u ^{2} + 9} - \frac{4}{- u ^{2} + 9}

= \int \frac{u}{- u ^{2} + 9} - \frac{4}{- 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{4}{- u ^{2} + 9} d u

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

\int \frac{4}{- u ^{2} + 9} d u = 4 arcsin (\frac{1}{3} u)

= - - u^{2} + 9 - 4 arcsin (\frac{1}{3} u)

Setze in

u = x + 4

ein

= - - (x + 4)^{2} + 9 - 4 arcsin (\frac{1}{3} (x + 4))

Multipliziere aus

- (x + 4)^{2} + 9 : - x^{2} - 8 x - 7

= - - x^{2} - 8 x - 7 - 4 arcsin (\frac{1}{3} (x + 4))

Füge eine Konstante zur Lösung hinzu

= - - x^{2} - 8 x - 7 - 4 arcsin (\frac{1}{3} (x + 4)) + C

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