What is the integral of 1/2 e^{sqrt(x)} ?

The integral of 1/2 e^{sqrt(x)} is e^{sqrt(x)}sqrt(x)-e^{sqrt(x)}+C

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integral of 1/2 e^{sqrt(x)}

\int \frac{1}{2} e^{x} d x

e^{x} x - e^{x} + C

Solution steps

\int \frac{1}{2} e^{x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int e^{x} d x

Apply u-substitution

= \frac{1}{2} \cdot \int e^{u} \cdot 2 u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \int e^{u} u d u

Apply Integration By Parts

= \frac{1}{2} \cdot 2 (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= \frac{1}{2} \cdot 2 (e^{u} u - e^{u})

Substitute back

u = x

= \frac{1}{2} \cdot 2 (e^{x} x - e^{x})

Simplify

\frac{1}{2} \cdot 2 (e^{x} x - e^{x}) : e^{x} x - e^{x}

= e^{x} x - e^{x}

Add a constant to the solution

= e^{x} x - e^{x} + C

What is the integral of 1/2 e^{sqrt(x)} ?
The integral of 1/2 e^{sqrt(x)} is e^{sqrt(x)}sqrt(x)-e^{sqrt(x)}+C