integral of cos(x)sqrt(1+sin(x))

Lösung

\int cos (x) 1 + sin (x) d x

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

Schritte zur Lösung

\int cos (x) 1 + sin (x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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