integral of (sin(x))/(2-sin(x))

Lösung

\int \frac{sin ( x )}{2 - sin ( x )} d x

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

Schritte zur Lösung

\int \frac{sin ( x )}{2 - sin ( x )} d x

\frac{sin ( x )}{2 - sin ( x )} = \frac{2}{2 - sin ( x )} - 1

= \int \frac{2}{2 - sin ( x )} - 1 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}{2 - sin ( x )} d x - \int 1 d x

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

\int 1 d x = x

= \frac{4}{3} arctan (\frac{1}{3} (2 tan (\frac{x}{2}) - 1)) - x

Füge eine Konstante zur Lösung hinzu

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

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