integral of ((x-5))/(x^2-18x+82)

Lösung

\int \frac{( x - 5 )}{x ^{2} - 18 x + 82} d x

\frac{1}{2} ln x^{2} - 18 x + 82 + 4 arctan (x - 9) + C

Schritte zur Lösung

\int \frac{x - 5}{x ^{2} - 18 x + 82} d x

Wende U-Substitution an

= \int \frac{u}{u ^{2} - 8 u + 17} d u

Vervollständige das Quadrat

u^{2} - 8 u + 17 : (u - 4)^{2} + 1

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

Wende U-Substitution an

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

Multipliziere aus

\frac{v + 4}{v ^{2} + 1} : \frac{v}{v ^{2} + 1} + \frac{4}{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{4}{v ^{2} + 1} d v

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

\int \frac{4}{v ^{2} + 1} d v = 4 arctan (v)

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

Ersetze zurück

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

Vereinfache

\frac{1}{2} ln (x - 5 - 4)^{2} + 1 + 4 arctan (x - 5 - 4) : \frac{1}{2} ln x^{2} - 18 x + 82 + 4 arctan (x - 9)

= \frac{1}{2} ln x^{2} - 18 x + 82 + 4 arctan (x - 9)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} - 18 x + 82 + 4 arctan (x - 9) + C

d er i v a t i v e f (x) = 1.5 x^{2} + 2 x^{1.5}

n = 1 \sum \infty \frac{1}{n ^{4} + n ^{2}}

\frac{d}{d x} (\frac{x ^{2} - 3 x - 4}{x + 1})

x \to \infty lim (\frac{3}{2 x})

\frac{d}{d x} (a x + \frac{a}{a x})