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

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

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integral of (e^{sqrt(x)})

\int (e^{x}) d x

2 (e^{x} x - e^{x}) + C

Solution steps

\int e^{x} d x

Apply u-substitution

= \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

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

Apply Integration By Parts

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

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

= 2 (e^{u} u - e^{u})

Substitute back

u = x

= 2 (e^{x} x - e^{x})

Add a constant to the solution

= 2 (e^{x} x - e^{x}) + C

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