integral of (3x+8)/(x(x-2)^2)

Lösung

\int \frac{3 x + 8}{x ( x - 2 ) ^{2}} d x

- \frac{7}{x - 2} + 2 ln ∣ x ∣ - 2 ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{3 x + 8}{x ( x - 2 ) ^{2}} d x

Multipliziere aus

\frac{3 x + 8}{x ( x - 2 ) ^{2}} : \frac{3 x}{x ( x - 2 ) ^{2}} + \frac{8}{x ( x - 2 ) ^{2}}

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{3 x}{x ( x - 2 ) ^{2}} d x + \int \frac{8}{x ( x - 2 ) ^{2}} d x

\int \frac{3 x}{x ( x - 2 ) ^{2}} d x = - \frac{3}{x - 2}

\int \frac{8}{x ( x - 2 ) ^{2}} d x = 8 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x - 2 ∣ - \frac{1}{2 ( x - 2 )})

= - \frac{3}{x - 2} + 8 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x - 2 ∣ - \frac{1}{2 ( x - 2 )})

Vereinfache

- \frac{3}{x - 2} + 8 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x - 2 ∣ - \frac{1}{2 ( x - 2 )}) : - \frac{7}{x - 2} + 2 ln ∣ x ∣ - 2 ln ∣ x - 2 ∣

= - \frac{7}{x - 2} + 2 ln ∣ x ∣ - 2 ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{7}{x - 2} + 2 ln ∣ x ∣ - 2 ln ∣ x - 2 ∣ + C

\frac{d}{d x} (\frac{2 x ^{2} - 1}{3 x + 5})

\int \frac{2 x + 3}{9 x ^{2} - 12 x + 8} d x

x \to \infty lim (\frac{1 - 5 e ^{2 x}}{3 e ^{x} + 4 e ^{2 x}})

\frac{d}{d x} (ln (csc (x) + cot (x)))

x \to \infty lim (- 2 x + 8)