integral of x/(sqrt(x^2-2ax+2a^2))

Lösung

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

x^{2} - 2 a x + 2 a^{2} + a ln (\frac{x - a + x ^{2} - 2 a x + 2 a ^{2}}{∣ a ∣}) + C

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} - 2 a x + 2 a^{2} : (x - a)^{2} + a^{2}

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

Wende U-Substitution an

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

Multipliziere aus

\frac{u + a}{u ^{2} + a ^{2}} : \frac{u}{u ^{2} + a ^{2}} + \frac{a}{u ^{2} + a ^{2}}

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

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

\int \frac{a}{u ^{2} + a ^{2}} d u = a ln (\frac{u + a ^{2} + u ^{2}}{∣ a ∣})

= u^{2} + a^{2} + a ln (\frac{u + a ^{2} + u ^{2}}{∣ a ∣})

Setze in

u = x - a

ein

= (x - a)^{2} + a^{2} + a ln \frac{x - a + a ^{2} + ( x - a ) ^{2}}{∣ a ∣}

Vereinfache

(x - a)^{2} + a^{2} + a ln \frac{x - a + a ^{2} + ( x - a ) ^{2}}{∣ a ∣} : x^{2} - 2 a x + 2 a^{2} + a ln (\frac{x - a + x ^{2} - 2 a x + 2 a ^{2}}{∣ a ∣})

= x^{2} - 2 a x + 2 a^{2} + a ln (\frac{x - a + x ^{2} - 2 a x + 2 a ^{2}}{∣ a ∣})

Füge eine Konstante zur Lösung hinzu

= x^{2} - 2 a x + 2 a^{2} + a ln (\frac{x - a + x ^{2} - 2 a x + 2 a ^{2}}{∣ a ∣}) + C

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