integral of (2-4x)/((x+2)(x-3))

Lösung

\int \frac{2 - 4 x}{( x + 2 ) ( x - 3 )} d x

- 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{2 - 4 x}{( x + 2 ) ( x - 3 )} d x

Multipliziere aus

\frac{2 - 4 x}{( x + 2 ) ( x - 3 )} : \frac{2}{( x + 2 ) ( x - 3 )} - \frac{4 x}{( x + 2 ) ( x - 3 )}

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 + 2 ) ( x - 3 )} d x - \int \frac{4 x}{( x + 2 ) ( x - 3 )} d x

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

\int \frac{4 x}{( x + 2 ) ( x - 3 )} d x = 4 (\frac{2}{5} ln ∣ x + 2 ∣ + \frac{3}{5} ln ∣ x - 3 ∣)

= 2 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - 4 (\frac{2}{5} ln ∣ x + 2 ∣ + \frac{3}{5} ln ∣ x - 3 ∣)

Vereinfache

2 (- \frac{1}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣) - 4 (\frac{2}{5} ln ∣ x + 2 ∣ + \frac{3}{5} ln ∣ x - 3 ∣) : - 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 3 ∣

= - 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= - 2 ln ∣ x + 2 ∣ - 2 ln ∣ x - 3 ∣ + C

\int - 2 x^{2} + 2 x + 4 d x

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

\int \frac{1}{x ^{2} - 10 x + 32} d x

\int sec^{2} (x) \cdot tan (x) d x

\int \frac{6 x}{( x ^{2} - 6 ) ^{5}} d x