What is the integral of 2^{1/2 x} ?

The integral of 2^{1/2 x} is (2^{1/2 (x+2)})/(ln(2))+C

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integral of 2^{1/2 x}

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

\frac{2 ^{\frac{1}{2} (x + 2)}}{ln ( 2 )} + C

Solution steps

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

Simplify

\frac{1}{2} x : \frac{x}{2}

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

Apply u-substitution

= \int 2^{u + 1} d u

2^{u + 1} = 2^{1} \cdot 2^{u}

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

Take the constant out:

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

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

Simplify

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

\int a^{x} d x = \frac{a ^{x}}{ln a}

= 2 \cdot \frac{2 ^{u}}{ln ( 2 )}

Substitute back

u = \frac{x}{2}

= 2 \cdot \frac{2 ^{\frac{x}{2}}}{ln ( 2 )}

Simplify

2 \cdot \frac{2 ^{\frac{x}{2}}}{ln ( 2 )} : \frac{2 ^{\frac{1}{2} (x + 2)}}{ln ( 2 )}

= \frac{2 ^{\frac{1}{2} (x + 2)}}{ln ( 2 )}

Add a constant to the solution

= \frac{2 ^{\frac{1}{2} (x + 2)}}{ln ( 2 )} + C

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