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

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

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integral of e^{sqrt(u)}

\int e^{u} d u

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

Solution steps

\int e^{u} d u

Apply u-substitution

: \int e^{v} \cdot 2 v d v

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

Take the constant out:

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

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

Apply Integration By Parts

: e^{v} v - \int e^{v} d v

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

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

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

Substitute back

v = u

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

Add a constant to the solution

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

\int \frac{( x - 1 ) ^{3}}{2 x} d x

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x \to + 1 + lim (1 - x^{2})

\int \frac{arctan ( 7 x )}{1 + 49 x ^{2}} d x

x \to \frac{π}{3} lim (1 - 2 cos (x))

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