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

Lösung

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

\frac{4}{x} + \frac{3}{2} ln ∣ x - 1 ∣ + ln ∣ x ∣ - \frac{5}{2} ln ∣ x + 1 ∣ + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{4}{x} - 2 ln ∣ x + 1 ∣ + 2 ln ∣ x - 1 ∣ - 1 + \frac{1}{2} x ln ∣ x + 1 ∣ - \frac{1}{2} x ln ∣ x - 1 ∣ - \int - \frac{1}{x} + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ d x

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

x \to 2 lim (2 x + c)

f (x) = \frac{1 - ln ( x )}{2 x ^{2}}

\frac{d y}{d x} = ∣ x - 2 ∣

t an g e n t f (x) = (x^{2} - 4)^{2}

in v erse l a pl a ce \frac{3}{( s + 1 ) ^{3}}