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

Lösung

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

- \frac{1}{32} ln ∣ x + 2 ∣ - \frac{5}{32} ln ∣ x - 2 ∣ + \frac{3}{16} ln ∣ x ∣ - \frac{1}{8 ( x + 2 )} - \frac{1}{8 ( x - 2 )} + \frac{3}{32 ( x + 2 )} - \frac{3}{32 ( x - 2 )} + C

Schritte zur Lösung

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

Multipliziere aus

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

\int \frac{2 x}{x ( x ^{2} - 4 ) ^{2}} d x = 2 (\frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )})

\int \frac{3}{x ( x ^{2} - 4 ) ^{2}} d x = 3 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 2 ∣ + \frac{1}{32 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{32 ( x - 2 )})

= 2 (\frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )}) + 3 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 2 ∣ + \frac{1}{32 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{32 ( x - 2 )})

Vereinfache

2 (\frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )}) + 3 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{32} ln ∣ x + 2 ∣ + \frac{1}{32 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{32 ( x - 2 )}) : - \frac{1}{32} ln ∣ x + 2 ∣ - \frac{5}{32} ln ∣ x - 2 ∣ + \frac{3}{16} ln ∣ x ∣ - \frac{1}{8 ( x + 2 )} - \frac{1}{8 ( x - 2 )} + \frac{3}{32 ( x + 2 )} - \frac{3}{32 ( x - 2 )}

= - \frac{1}{32} ln ∣ x + 2 ∣ - \frac{5}{32} ln ∣ x - 2 ∣ + \frac{3}{16} ln ∣ x ∣ - \frac{1}{8 ( x + 2 )} - \frac{1}{8 ( x - 2 )} + \frac{3}{32 ( x + 2 )} - \frac{3}{32 ( x - 2 )}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{32} ln ∣ x + 2 ∣ - \frac{5}{32} ln ∣ x - 2 ∣ + \frac{3}{16} ln ∣ x ∣ - \frac{1}{8 ( x + 2 )} - \frac{1}{8 ( x - 2 )} + \frac{3}{32 ( x + 2 )} - \frac{3}{32 ( x - 2 )} + C