What is the integral of e^{2sqrt(t)} ?

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

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integral of e^{2sqrt(t)}

\int e^{2 t} d t

e^{2 t} t - \frac{1}{2} e^{2 t} + C

Solution steps

\int e^{2 t} d t

Apply u-substitution

: \int \frac{e ^{u} u}{2} d u

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = 2 t

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

Simplify

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

= e^{2 t} t - \frac{1}{2} e^{2 t}

Add a constant to the solution

= e^{2 t} t - \frac{1}{2} e^{2 t} + C

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