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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

= \int \frac{x}{x ^{4} - 4} - \frac{3}{x x ^{4} - 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{x}{x ^{4} - 4} d x - \int \frac{3}{x x ^{4} - 4} d x

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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