integral of (x+9)/(x^2+8x+20)

Lösung

\int \frac{x + 9}{x ^{2} + 8 x + 20} d x

\frac{1}{2} ln x^{2} + 8 x + 20 + \frac{5}{2} arctan (\frac{x + 4}{2}) + C

Schritte zur Lösung

\int \frac{x + 9}{x ^{2} + 8 x + 20} d x

Wende U-Substitution an

= \int \frac{u}{u ^{2} - 10 u + 29} d u

Vervollständige das Quadrat

u^{2} - 10 u + 29 : (u - 5)^{2} + 4

= \int \frac{u}{( u - 5 ) ^{2} + 4} d u

Wende U-Substitution an

= \int \frac{v + 5}{v ^{2} + 4} d v

Multipliziere aus

\frac{v + 5}{v ^{2} + 4} : \frac{v}{v ^{2} + 4} + \frac{5}{v ^{2} + 4}

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{v}{v ^{2} + 4} d v + \int \frac{5}{v ^{2} + 4} d v

\int \frac{v}{v ^{2} + 4} d v = \frac{1}{2} ln v^{2} + 4

\int \frac{5}{v ^{2} + 4} d v = \frac{5}{2} arctan (\frac{v}{2})

= \frac{1}{2} ln v^{2} + 4 + \frac{5}{2} arctan (\frac{v}{2})

Ersetze zurück

= \frac{1}{2} ln (x + 9 - 5)^{2} + 4 + \frac{5}{2} arctan (\frac{x + 9 - 5}{2})

Vereinfache

\frac{1}{2} ln (x + 9 - 5)^{2} + 4 + \frac{5}{2} arctan (\frac{x + 9 - 5}{2}) : \frac{1}{2} ln x^{2} + 8 x + 20 + \frac{5}{2} arctan (\frac{x + 4}{2})

= \frac{1}{2} ln x^{2} + 8 x + 20 + \frac{5}{2} arctan (\frac{x + 4}{2})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} + 8 x + 20 + \frac{5}{2} arctan (\frac{x + 4}{2}) + C

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