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Beliebt Rechnen >

integral of x/(x^2+10x+50)

Lösung

\int \frac{x}{x ^{2} + 10 x + 50} d x

Lösung

\frac{1}{2} ln x^{2} + 10 x + 50 - arctan (\frac{1}{5} (x + 5)) + C

Schritte zur Lösung

\int \frac{x}{x ^{2} + 10 x + 50} d x

Vervollständige das Quadrat

x^{2} + 10 x + 50 : (x + 5)^{2} + 25

= \int \frac{x}{( x + 5 ) ^{2} + 25} d x

Wende U-Substitution an

= \int \frac{u - 5}{u ^{2} + 25} d u

Multipliziere aus

\frac{u - 5}{u ^{2} + 25} : \frac{u}{u ^{2} + 25} - \frac{5}{u ^{2} + 25}

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} + 25} d u - \int \frac{5}{u ^{2} + 25} d u

\int \frac{u}{u ^{2} + 25} d u = \frac{1}{2} ln u^{2} + 25

\int \frac{5}{u ^{2} + 25} d u = arctan (\frac{u}{5})

= \frac{1}{2} ln u^{2} + 25 - arctan (\frac{u}{5})

Setze in

u = x + 5

ein

= \frac{1}{2} ln (x + 5)^{2} + 25 - arctan (\frac{x + 5}{5})

Vereinfache

\frac{1}{2} ln (x + 5)^{2} + 25 - arctan (\frac{x + 5}{5}) : \frac{1}{2} ln x^{2} + 10 x + 50 - arctan (\frac{1}{5} (x + 5))

= \frac{1}{2} ln x^{2} + 10 x + 50 - arctan (\frac{1}{5} (x + 5))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 10 x + 50 - arctan (\frac{1}{5} (x + 5)) + C

Graph

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