integral of ((x-3))/((x-2)(x^2+1))

Lösung

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

- \frac{1}{5} ln ∣ x - 2 ∣ + \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 7 arctan (x)) + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x - 3}{( x - 2 ) ( x ^{2} + 1 )} : - \frac{1}{5 ( x - 2 )} + \frac{x + 7}{5 ( x ^{2} + 1 )}

= \int - \frac{1}{5 ( x - 2 )} + \frac{x + 7}{5 ( x ^{2} + 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{1}{5 ( x - 2 )} d x + \int \frac{x + 7}{5 ( x ^{2} + 1 )} d x

\int \frac{1}{5 ( x - 2 )} d x = \frac{1}{5} ln ∣ x - 2 ∣

\int \frac{x + 7}{5 ( x ^{2} + 1 )} d x = \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 7 arctan (x))

= - \frac{1}{5} ln ∣ x - 2 ∣ + \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 7 arctan (x))

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{5} ln ∣ x - 2 ∣ + \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 7 arctan (x)) + C

\int 3 x (3 x + 2) (4 x - 1) d x

\int - 1 x d x

\int \frac{( x ^{2} + 5 )}{2 x} d x

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

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