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

Lösung

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

\frac{x ^{3}}{3} + 2 x - 2 ln \frac{x}{2} + 1 + 2 ln \frac{x}{2} - 1 + C

Schritte zur Lösung

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

Lange Division

\frac{x ^{4}}{x ^{2} - 2} : x^{2} + 2 + \frac{4}{x ^{2} - 2}

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

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

\int 2 d x = 2 x

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

= \frac{x ^{3}}{3} + 2 x - 2 ln \frac{x}{2} + 1 + 2 ln \frac{x}{2} - 1

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{3} + 2 x - 2 ln \frac{x}{2} + 1 + 2 ln \frac{x}{2} - 1 + C

\int cos^{7} (x) sin^{4} (x) d x

\int \frac{1 - e ^{- x}}{e ^{3 x}} d x

\int 162 + 162 cos (x) d x

\int \frac{3 x - 2}{x ^{2} + 3 x - 10} d x

\int \frac{2 x + 3}{( x ^{2} + 3 x - 8 ) ^{2}} d x