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Beliebt Rechnen >

integral of x/(sqrt(x^2+4x+8))

Lösung

\int \frac{x}{x ^{2} + 4 x + 8} d x

Lösung

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

Schritte zur Lösung

\int \frac{x}{x ^{2} + 4 x + 8} d x

Vervollständige das Quadrat

x^{2} + 4 x + 8 : (x + 2)^{2} + 4

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

\int \frac{2}{u ^{2} + 4} d u = 2 ln (\frac{1}{2} u + 4 + u^{2})

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

Setze in

u = x + 2

ein

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

Vereinfache

(x + 2)^{2} + 4 - 2 ln (\frac{1}{2} x + 2 + 4 + (x + 2)^{2}) : x^{2} + 4 x + 8 - 2 ln (\frac{1}{2} x + 2 + x^{2} + 4 x + 8)

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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