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

Lösung

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

- ln ∣ 3 - x ∣ + 2 (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x - 3 ∣) + C

Schritte zur Lösung

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

Multipliziere aus

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

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

\int \frac{x}{x ( 3 - x )} d x = - ln ∣ 3 - x ∣

\int \frac{2}{x ( 3 - x )} d x = 2 (\frac{1}{3} ln ∣ x ∣ - \frac{1}{3} ln ∣ x - 3 ∣)

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

Füge eine Konstante zur Lösung hinzu

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

\int 5 (cos (x)) (sin^{4} (x)) d x

\int sin^{4} (x) sin (x) d x

\int 2 + 4 - x d x

\int 5 sinh (ln (2) - \frac{x}{3}) d x

\int \frac{sin ( 2 x )}{e ^{- 4 x}} d x