integral of (2x+1)/(x(x-2))

Lösung

\int \frac{2 x + 1}{x ( x - 2 )} d x

- \frac{1}{2} ln ∣ x ∣ + \frac{5}{2} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{2 x + 1}{x ( x - 2 )} d x

Multipliziere aus

\frac{2 x + 1}{x ( x - 2 )} : \frac{2 x}{x ( x - 2 )} + \frac{1}{x ( x - 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{2 x}{x ( x - 2 )} d x + \int \frac{1}{x ( x - 2 )} d x

\int \frac{2 x}{x ( x - 2 )} d x = 2 ln ∣ x - 2 ∣

\int \frac{1}{x ( x - 2 )} d x = - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣

= 2 ln ∣ x - 2 ∣ - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣

Vereinfache

2 ln ∣ x - 2 ∣ - \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2 ∣ : - \frac{1}{2} ln ∣ x ∣ + \frac{5}{2} ln ∣ x - 2 ∣

= - \frac{1}{2} ln ∣ x ∣ + \frac{5}{2} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln ∣ x ∣ + \frac{5}{2} ln ∣ x - 2 ∣ + C

\int (4 x^{2} + 9)^{2} d x

\int 3 cos (x) + 2 sin (x) d x

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

\int 1 - \frac{1}{x ^{2} + 1} d x

in t e g r a l 5