Evaluate
`int 2^(log_(4)x)dx`

Answer

To evaluate the integral \( \int 2^{\log_{4} x} \, dx \), we can follow these steps:

### Step 1: Rewrite the exponent using logarithmic identities
We know that \( \log_{b} a = \frac{\log_{k} a}{\log_{k} b} \) for any base \( k \). Therefore, we can rewrite \( \log_{4} x \) in terms of base 2:
\[ \log_{4} x = \frac{\log_{2} x}{\log_{2} 4} \]
Since \( \log_{2} 4 = 2 \), we have:
\[ \log_{4} x = \frac{\log_{2} x}{2} \]
Thus,
\[ 2^{\log_{4} x} = 2^{\frac{\log_{2} x}{2}} = (2^{\log_{2} x})^{\frac{1}{2}} = x^{\frac{1}{2}} = \sqrt{x} \]

### Step 2: Rewrite the integral
Now, we can rewrite the integral as:
\[ \int 2^{\log_{4} x} \, dx = \int \sqrt{x} \, dx \]

### Step 3: Integrate
The integral of \( \sqrt{x} \) can be calculated as follows:
\[ \int \sqrt{x} \, dx = \int x^{\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3} x^{\frac{3}{2}} + C \]

### Final Answer
Thus, the evaluated integral is:
\[ \int 2^{\log_{4} x} \, dx = \frac{2}{3} x^{\frac{3}{2}} + C \]

---