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

Lösung

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

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

= 2 arcsin (\frac{1}{2} x) - \frac{3 4 - x ^{2}}{4 x}

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l 2 x^{- 1}

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