integral of (x^5)/((x^2+4)^2)

Lösung

\int \frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} d x

\frac{x ^{2}}{2} - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4} + C

Schritte zur Lösung

\int \frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} d x

Lange Division

\frac{x ^{5}}{( x ^{2} + 4 ) ^{2}} : x + \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16}

= \int x + \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} 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 d x + \int \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} d x

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

\int \frac{- 8 x ^{3} - 16 x}{x ^{4} + 8 x ^{2} + 16} d x = - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4}

= \frac{x ^{2}}{2} - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{2}}{2} - \frac{1}{4} x^{3} sin (2 arctan (\frac{x}{2})) + x^{2} - 4 ln x^{2} + 4 + 4 + \frac{8}{x ^{2} + 4} + C

\frac{d}{d x} (6 x^{3} - 4 x^{2} + 5 x)

t an g e n t f (x) = \frac{9 x}{x + 4}, at x = 8

\int - \frac{1}{4} sin (2 x) d x

d er i v a t i v e f (x) = \frac{6 x}{x + 6}

\frac{d}{d x} (ln (3 x^{2} + 5))