integral of x/(sqrt(x^2+4x-5))

Lösung

\int \frac{x}{x ^{2} + 4 x - 5} d x

x^{2} + 4 x - 5 - 2 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2) + C

Schritte zur Lösung

\int \frac{x}{x ^{2} + 4 x - 5} d x

Vervollständige das Quadrat

x^{2} + 4 x - 5 : (x + 2)^{2} - 9

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

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

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

Setze in

u = x + 2

ein

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

Vereinfache

(x + 2)^{2} - 9 - 2 ln (\frac{1}{3} (x + 2)^{2} - 9 + x + 2) : x^{2} + 4 x - 5 - 2 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2)

= x^{2} + 4 x - 5 - 2 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2)

Füge eine Konstante zur Lösung hinzu

= x^{2} + 4 x - 5 - 2 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2) + C

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