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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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