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

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

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integral of sqrt(e^x)

\int e^{x} d x

2 e^{\frac{x}{2}} + C

Solution steps

\int e^{x} d x

Apply radical rule:

assuming

a \geq 0

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

Apply u-substitution

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

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

Use the common integral:

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

= 2 e^{u}

Substitute back

u = \frac{x}{2}

= 2 e^{\frac{x}{2}}

Add a constant to the solution

= 2 e^{\frac{x}{2}} + C

x \to 0 - lim (\frac{x ^{2}}{x})

y = (4 x - 1)^{3}

y = 7 x - x^{2}, at x = 1

f (x) = 3^{c o s (x)}

\int tan (x) d x