integral of ((x+6)/(x^2+11x+18))

Lösung

\int (\frac{x + 6}{x ^{2} + 11 x + 18}) d x

2 (\frac{1}{4} ln 4 x^{2} + 44 x + 72 - \frac{1}{28} (ln \frac{2 x + 11}{7} + 1 - ln \frac{2 x + 11}{7} - 1)) + C

Schritte zur Lösung

\int \frac{x + 6}{x ^{2} + 11 x + 18} d x

Wende U-Substitution an

= \int \frac{u}{u ^{2} - u - 12} d u

Vervollständige das Quadrat

u^{2} - u - 12 : (u - \frac{1}{2})^{2} - \frac{49}{4}

= \int \frac{u}{( u - \frac{1}{2} ) ^{2} - \frac{49}{4}} d u

Wende U-Substitution an

= \int \frac{2 ( 2 v + 1 )}{4 v ^{2} - 49} d v

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{2 v + 1}{4 v ^{2} - 49} d v

Multipliziere aus

\frac{2 v + 1}{4 v ^{2} - 49} : \frac{2 v}{4 v ^{2} - 49} + \frac{1}{4 v ^{2} - 49}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{2 v}{4 v ^{2} - 49} d v + \int \frac{1}{4 v ^{2} - 49} d v)

\int \frac{2 v}{4 v ^{2} - 49} d v = \frac{1}{4} ln 4 v^{2} - 49

\int \frac{1}{4 v ^{2} - 49} d v = - \frac{1}{28} (ln \frac{2}{7} v + 1 - ln \frac{2}{7} v - 1)

= 2 (\frac{1}{4} ln 4 v^{2} - 49 - \frac{1}{28} (ln \frac{2}{7} v + 1 - ln \frac{2}{7} v - 1))

Ersetze zurück

= 2 (\frac{1}{4} ln 4 (x + 6 - \frac{1}{2})^{2} - 49 - \frac{1}{28} (ln \frac{2}{7} (x + 6 - \frac{1}{2}) + 1 - ln \frac{2}{7} (x + 6 - \frac{1}{2}) - 1))

Vereinfache

2 (\frac{1}{4} ln 4 (x + 6 - \frac{1}{2})^{2} - 49 - \frac{1}{28} (ln \frac{2}{7} (x + 6 - \frac{1}{2}) + 1 - ln \frac{2}{7} (x + 6 - \frac{1}{2}) - 1)) : 2 (\frac{1}{4} ln 4 x^{2} + 44 x + 72 - \frac{1}{28} (ln \frac{2 x + 11}{7} + 1 - ln \frac{2 x + 11}{7} - 1))

= 2 (\frac{1}{4} ln 4 x^{2} + 44 x + 72 - \frac{1}{28} (ln \frac{2 x + 11}{7} + 1 - ln \frac{2 x + 11}{7} - 1))

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{1}{4} ln 4 x^{2} + 44 x + 72 - \frac{1}{28} (ln \frac{2 x + 11}{7} + 1 - ln \frac{2 x + 11}{7} - 1)) + C

\frac{d}{d x} (x^{6} + 3 x^{4} + 7 x^{2})

l a pl a ce t r an s f or m cos (π t)

x \to 0 lim ((x^{2}) cos (\frac{8}{x}))

\frac{\partial}{\partial x} (- 10 x y sin (x y^{2}))

d er i v a t i v e f (x) = (2 x - x^{3}) (3 x + 7)