integral of (3-x)/((3x+1)(x-1))

Lösung

\int \frac{3 - x}{( 3 x + 1 ) ( x - 1 )} d x

- \frac{5}{6} ln ∣ 3 x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣ + C

Schritte zur Lösung

\int \frac{3 - x}{( 3 x + 1 ) ( x - 1 )} d x

Ermittle den Partialbruch von

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

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

\int \frac{5}{2 ( 3 x + 1 )} d x = \frac{5}{6} ln ∣ 3 x + 1 ∣

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

= - \frac{5}{6} ln ∣ 3 x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{5}{6} ln ∣ 3 x + 1 ∣ + \frac{1}{2} ln ∣ x - 1 ∣ + C

\int arcsin (sin (x)) d x

\int \frac{2}{x 9 x ^{2} - 25} d x

\int \frac{1}{z ^{2} z ^{2} + a ^{2}} d z

\int sin^{4} (θ) cos^{4} (θ) d θ

\int \frac{2 x}{4 + 3 x ^{2}} d x