integral of x/(sqrt(x^2+20x+110))

Lösung

\int \frac{x}{x ^{2} + 20 x + 110} d x

x^{2} + 20 x + 110 - 10 ln \frac{1}{10} (x + 10) + \frac{1}{10} (x^{2} + 20 x + 110) + C

Schritte zur Lösung

\int \frac{x}{x ^{2} + 20 x + 110} d x

Vervollständige das Quadrat

x^{2} + 20 x + 110 : (x + 10)^{2} + 10

= \int \frac{x}{( x + 10 ) ^{2} + 10} d x

Wende U-Substitution an

= \int \frac{u - 10}{u ^{2} + 10} d u

Multipliziere aus

\frac{u - 10}{u ^{2} + 10} : \frac{u}{u ^{2} + 10} - \frac{10}{u ^{2} + 10}

= \int \frac{u}{u ^{2} + 10} - \frac{10}{u ^{2} + 10} 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} + 10} d u - \int \frac{10}{u ^{2} + 10} d u

\int \frac{u}{u ^{2} + 10} d u = u^{2} + 10

\int \frac{10}{u ^{2} + 10} d u = 10 ln \frac{1}{10} u + \frac{1}{10} (10 + u^{2})

= u^{2} + 10 - 10 ln \frac{1}{10} u + \frac{1}{10} (10 + u^{2})

Setze in

u = x + 10

ein

= (x + 10)^{2} + 10 - 10 ln \frac{1}{10} (x + 10) + \frac{1}{10} (10 + (x + 10)^{2})

Vereinfache

(x + 10)^{2} + 10 - 10 ln \frac{1}{10} (x + 10) + \frac{1}{10} (10 + (x + 10)^{2}) : x^{2} + 20 x + 110 - 10 ln \frac{1}{10} (x + 10) + \frac{1}{10} (x^{2} + 20 x + 110)

= x^{2} + 20 x + 110 - 10 ln \frac{1}{10} (x + 10) + \frac{1}{10} (x^{2} + 20 x + 110)

Füge eine Konstante zur Lösung hinzu

= x^{2} + 20 x + 110 - 10 ln \frac{1}{10} (x + 10) + \frac{1}{10} (x^{2} + 20 x + 110) + C

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