integral of (22x+88)/(2x^2+9x-5)

Lösung

\int \frac{22 x + 88}{2 x ^{2} + 9 x - 5} d x

2 ln ∣ 22 x + 110 ∣ + 9 ln ∣ 22 x - 11 ∣ + C

Schritte zur Lösung

\int \frac{22 x + 88}{2 x ^{2} + 9 x - 5} d x

Wende U-Substitution an

= \int \frac{11 u}{u ^{2} - 77 u - 2178} d u

Entferne die Konstante:

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

= 11 \cdot \int \frac{u}{u ^{2} - 77 u - 2178} d u

Ermittle den Partialbruch von

\frac{u}{u ^{2} - 77 u - 2178} : \frac{2}{11 ( u + 22 )} + \frac{9}{11 ( u - 99 )}

= 11 \cdot \int \frac{2}{11 ( u + 22 )} + \frac{9}{11 ( u - 99 )} d u

Wende die Summenregel an:

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

= 11 (\int \frac{2}{11 ( u + 22 )} d u + \int \frac{9}{11 ( u - 99 )} d u)

\int \frac{2}{11 ( u + 22 )} d u = \frac{2}{11} ln ∣ u + 22 ∣

\int \frac{9}{11 ( u - 99 )} d u = \frac{9}{11} ln ∣ u - 99 ∣

= 11 (\frac{2}{11} ln ∣ u + 22 ∣ + \frac{9}{11} ln ∣ u - 99 ∣)

Setze in

u = 22 x + 88

ein

= 11 (\frac{2}{11} ln ∣ 22 x + 88 + 22 ∣ + \frac{9}{11} ln ∣ 22 x + 88 - 99 ∣)

Vereinfache

11 (\frac{2}{11} ln ∣ 22 x + 88 + 22 ∣ + \frac{9}{11} ln ∣ 22 x + 88 - 99 ∣) : 2 ln ∣ 22 x + 110 ∣ + 9 ln ∣ 22 x - 11 ∣

= 2 ln ∣ 22 x + 110 ∣ + 9 ln ∣ 22 x - 11 ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ 22 x + 110 ∣ + 9 ln ∣ 22 x - 11 ∣ + C

\int u^{- \frac{3}{2}} d u

d er i v a t i v e y = csc^{2} (x)

\int x (x + 5) d x

\frac{\partial}{\partial x} (2 x + y^{2})

in v erse l a pl a ce \frac{10}{s} (\frac{1}{10 s + 1})