integral of 2/(sqrt(5-5x^2))

Lösung

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

2 (\frac{x}{5 - 5 x ^{2}} - \frac{1}{5} (- arcsin (x) + \frac{x}{1 - x ^{2}})) + C

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{5 - 5 x ^{2}} d x

Wende die partielle Integration an

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

\int \frac{5 x ^{2}}{( 5 - 5 x ^{2} ) ^{\frac{3}{2}}} d x = \frac{1}{5} (- arcsin (x) + \frac{x}{1 - x ^{2}})

= 2 (\frac{x}{5 - 5 x ^{2}} - \frac{1}{5} (- arcsin (x) + \frac{x}{1 - x ^{2}}))

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{x}{5 - 5 x ^{2}} - \frac{1}{5} (- arcsin (x) + \frac{x}{1 - x ^{2}})) + C

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