integral of cos(y)sqrt(e^y)

Lösung

\int cos (y) e^{y} d y

\frac{2}{5} e^{\frac{y}{2}} cos (y) + \frac{4}{5} e^{\frac{y}{2}} sin (y) + C

Schritte zur Lösung

\int cos (y) e^{y} d y

Wende die partielle Integration an

= e^{\frac{y}{2}} sin (y) - \int \frac{1}{2} e^{\frac{y}{2}} sin (y) d y

\int \frac{1}{2} e^{\frac{y}{2}} sin (y) d y = - \frac{2}{5} e^{\frac{y}{2}} cos (y) + \frac{1}{5} e^{\frac{y}{2}} sin (y)

= e^{\frac{y}{2}} sin (y) - (- \frac{2}{5} e^{\frac{y}{2}} cos (y) + \frac{1}{5} e^{\frac{y}{2}} sin (y))

Vereinfache

e^{\frac{y}{2}} sin (y) - (- \frac{2}{5} e^{\frac{y}{2}} cos (y) + \frac{1}{5} e^{\frac{y}{2}} sin (y)) : \frac{2}{5} e^{\frac{y}{2}} cos (y) + \frac{4}{5} e^{\frac{y}{2}} sin (y)

= \frac{2}{5} e^{\frac{y}{2}} cos (y) + \frac{4}{5} e^{\frac{y}{2}} sin (y)

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} e^{\frac{y}{2}} cos (y) + \frac{4}{5} e^{\frac{y}{2}} sin (y) + C

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