integral of (x-7)/(x^2-18x+82)

Lösung

\int \frac{x - 7}{x ^{2} - 18 x + 82} d x

\frac{1}{2} ln x^{2} - 18 x + 82 + 2 arctan (x - 9) + C

Schritte zur Lösung

\int \frac{x - 7}{x ^{2} - 18 x + 82} d x

Wende U-Substitution an

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

Vervollständige das Quadrat

u^{2} - 4 u + 5 : (u - 2)^{2} + 1

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

Wende U-Substitution an

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

Multipliziere aus

\frac{v + 2}{v ^{2} + 1} : \frac{v}{v ^{2} + 1} + \frac{2}{v ^{2} + 1}

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} + 1} d v + \int \frac{2}{v ^{2} + 1} d v

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

\int \frac{2}{v ^{2} + 1} d v = 2 arctan (v)

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

Ersetze zurück

= \frac{1}{2} ln (x - 7 - 2)^{2} + 1 + 2 arctan (x - 7 - 2)

Vereinfache

\frac{1}{2} ln (x - 7 - 2)^{2} + 1 + 2 arctan (x - 7 - 2) : \frac{1}{2} ln x^{2} - 18 x + 82 + 2 arctan (x - 9)

= \frac{1}{2} ln x^{2} - 18 x + 82 + 2 arctan (x - 9)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} - 18 x + 82 + 2 arctan (x - 9) + C

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