integral of (-x^2+2x-8)/((x^2+4)(x-2))

Lösung

\int \frac{- x ^{2} + 2 x - 8}{( x ^{2} + 4 ) ( x - 2 )} d x

\frac{1}{4} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2}) - \frac{1}{4} ln x^{2} + 4 - ln \frac{x}{2} - 1 + C

Schritte zur Lösung

\int \frac{- x ^{2} + 2 x - 8}{( x ^{2} + 4 ) ( x - 2 )} d x

Multipliziere aus

\frac{- x ^{2} + 2 x - 8}{( x ^{2} + 4 ) ( x - 2 )} : - \frac{x ^{2}}{( x ^{2} + 4 ) ( x - 2 )} + \frac{2 x}{( x ^{2} + 4 ) ( x - 2 )} - \frac{8}{( x ^{2} + 4 ) ( x - 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{x ^{2}}{( x ^{2} + 4 ) ( x - 2 )} d x + \int \frac{2 x}{( x ^{2} + 4 ) ( x - 2 )} d x - \int \frac{8}{( x ^{2} + 4 ) ( x - 2 )} d x

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

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

\int \frac{8}{( x ^{2} + 4 ) ( x - 2 )} d x = - \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2}) + ln \frac{x}{2} - 1

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

Vereinfache

- (\frac{1}{2} (\frac{1}{2} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2})) + \frac{1}{2} ln ∣ x - 2 ∣) + 2 (\frac{1}{4} (- \frac{1}{2} ln x^{2} + 4 + arctan (\frac{x}{2})) + \frac{1}{4} ln ∣ x - 2 ∣) - (- \frac{1}{2} ln \frac{x ^{2}}{4} + 1 - arctan (\frac{x}{2}) + ln \frac{x}{2} - 1) : \frac{1}{4} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2}) - \frac{1}{4} ln x^{2} + 4 - ln \frac{x}{2} - 1

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} ln \frac{x ^{2}}{4} + 1 + arctan (\frac{x}{2}) - \frac{1}{4} ln x^{2} + 4 - ln \frac{x}{2} - 1 + C

\int (x^{2} + 10 x) cos (x) d x

\frac{d}{d x} (arcsinh (x^{2}))

\int \frac{2 x}{x ^{2} - 6} d x

t a y l or ln (x), 8

d er i v a t i v e \frac{1}{3} arctan (\frac{x}{3})