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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{x ^{2} + 5}{x ^{3} - x ^{2} + x + 3} : \frac{1}{x + 1} + \frac{2}{x ^{2} - 2 x + 3}

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

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (e^{-sx})

\int (e^{- s x}) d x

tangent y=sin(sin(x)),(4pi,0)

t an g e n t y = sin (sin (x)), (4 π, 0)

limit as x approaches 0 of x^{5sin(x)}

x \to 0 lim (x^{5 s i n (x)})

limit as x approaches infinity of 3/5

x \to \infty lim (\frac{3}{5})

(dx}{dy}=\frac{x^2)/y

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