What is the integral of cos(y)sqrt(e^y) ?

The integral of cos(y)sqrt(e^y) is 2/5 e^{y/2}cos(y)+4/5 e^{y/2}sin(y)+C

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

\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

Solution steps

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

Apply Integration By Parts

= 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))

Simplify

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)

Add a constant to the solution

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

What is the integral of cos(y)sqrt(e^y) ?
The integral of cos(y)sqrt(e^y) is 2/5 e^{y/2}cos(y)+4/5 e^{y/2}sin(y)+C