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Beliebt Rechnen >

integral of (x^4-x^3+2x^2-2x-8)/(x^2+4)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{x ^{4} - x ^{3} + 2 x ^{2} - 2 x - 8}{x ^{2} + 4} : \frac{x ^{4}}{x ^{2} + 4} - \frac{x ^{3}}{x ^{2} + 4} + \frac{2 x ^{2}}{x ^{2} + 4} - \frac{2 x}{x ^{2} + 4} - \frac{8}{x ^{2} + 4}

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

\int \frac{x ^{4}}{x ^{2} + 4} d x = 8 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2}))

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

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

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

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

= 8 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2})) - \frac{1}{2} (x^{2} + 4 - 4 ln x^{2} + 4) + 4 (- arctan (\frac{x}{2}) + \frac{x}{2}) - ln x^{2} + 4 - 4 arctan (\frac{x}{2})

Vereinfache

8 (\frac{x ^{3}}{24} - \frac{x}{2} + arctan (\frac{x}{2})) - \frac{1}{2} (x^{2} + 4 - 4 ln x^{2} + 4) + 4 (- arctan (\frac{x}{2}) + \frac{x}{2}) - ln x^{2} + 4 - 4 arctan (\frac{x}{2}) : \frac{x ^{3}}{3} - \frac{1}{2} x^{2} - 2 x + ln x^{2} + 4 - 2

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

limit as x approaches 4 of x/(x^2-3x)

x \to 4 lim (\frac{x}{x ^{2} - 3 x})

y^{^{'}} = (x + 1) y^{2}

integral of (x^3)/(x^2+5)

\int \frac{x ^{3}}{x ^{2} + 5} d x

derivative of (dy/(dx))=4x

\frac{d}{d x} (\frac{d y}{d x}) = 4 x

tangent f(x)=-3e^{-2x}+3

t an g e n t f (x) = - 3 e^{- 2 x} + 3