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Beliebt Rechnen >

integral of x/(sqrt(x^2-20x+84))

Lösung

\int \frac{x}{x ^{2} - 20 x + 84} d x

Lösung

x^{2} - 20 x + 84 + 10 ln (\frac{1}{4} x^{2} - 20 x + 84 + x - 10) + C

Schritte zur Lösung

\int \frac{x}{x ^{2} - 20 x + 84} d x

Vervollständige das Quadrat

x^{2} - 20 x + 84 : (x - 10)^{2} - 16

= \int \frac{x}{( x - 10 ) ^{2} - 16} d x

Wende U-Substitution an

= \int \frac{u + 10}{u ^{2} - 16} d u

Multipliziere aus

\frac{u + 10}{u ^{2} - 16} : \frac{u}{u ^{2} - 16} + \frac{10}{u ^{2} - 16}

= \int \frac{u}{u ^{2} - 16} + \frac{10}{u ^{2} - 16} 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} - 16} d u + \int \frac{10}{u ^{2} - 16} d u

\int \frac{u}{u ^{2} - 16} d u = u^{2} - 16

\int \frac{10}{u ^{2} - 16} d u = 10 ln (\frac{1}{4} u^{2} - 16 + u)

= u^{2} - 16 + 10 ln (\frac{1}{4} u^{2} - 16 + u)

Setze in

u = x - 10

ein

= (x - 10)^{2} - 16 + 10 ln (\frac{1}{4} (x - 10)^{2} - 16 + x - 10)

Vereinfache

(x - 10)^{2} - 16 + 10 ln (\frac{1}{4} (x - 10)^{2} - 16 + x - 10) : x^{2} - 20 x + 84 + 10 ln (\frac{1}{4} x^{2} - 20 x + 84 + x - 10)

= x^{2} - 20 x + 84 + 10 ln (\frac{1}{4} x^{2} - 20 x + 84 + x - 10)

Füge eine Konstante zur Lösung hinzu

= x^{2} - 20 x + 84 + 10 ln (\frac{1}{4} x^{2} - 20 x + 84 + x - 10) + C

Graph

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