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

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

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

\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

Solution steps

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

Apply Integration By Parts

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

= 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))

Simplify

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)

Add a constant to the solution

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

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