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

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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