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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of in^2x

\int i n^{2} x d x

tangent y=sqrt(x^2-x+10),\at x=3

t an g e n t y = x^{2} - x + 10, at x = 3

derivative of cos(xln(cos(x))+xsin(x))

\frac{d}{d x} (cos (x) ln (cos (x)) + x sin (x))

derivative of (ln(4x)/(4x))

\frac{d}{d x} (\frac{ln ( 4 x )}{4 x})

fläche y=4-x^2,y=x+2

a re a y = 4 - x^{2}, y = x + 2