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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

limit as x approaches 2 of (x^3+x^2-8x-12)/(x^4+4x^3+5x^2+4x+4)

x \to 2 lim (\frac{x ^{3} + x ^{2} - 8 x - 12}{x ^{4} + 4 x ^{3} + 5 x ^{2} + 4 x + 4})

y^'=((t^4+y^4))/(ty^3)

y^{^{'}} = \frac{( t ^{4} + y ^{4} )}{t y ^{3}}

limit as x approaches+0 of x

x \to + 0 lim (x)

integral of (2x+3)^4

\int (2 x + 3)^{4} d x

integral of e^{uv}

\int e^{uv} d u