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

Answer

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

### Step 1: Simplify the expression \( 2^{\log_{4} x} \)

We know that:
\[ \log_{4} x = \frac{\log_{2} x}{\log_{2} 4} \]
Since \( \log_{2} 4 = 2 \), we can rewrite it as:
\[ \log_{4} x = \frac{\log_{2} x}{2} \]
Thus, we have:
\[ 2^{\log_{4} x} = 2^{\frac{\log_{2} x}{2}} = 2^{\log_{2} x^{1/2}} = x^{1/2} \]

### Step 2: Rewrite the integral

Now we can rewrite the integral:
\[ I = \int 2^{\log_{4} x} \, dx = \int x^{1/2} \, dx \]

### Step 3: Integrate \( x^{1/2} \)

The integral of \( x^{1/2} \) can be computed using the power rule:
\[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]
For \( n = \frac{1}{2} \):
\[ \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} + C = \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C \]

### Step 4: Write the final answer

Thus, the final result of the integral is:
\[ I = \frac{2}{3} x^{3/2} + C \]

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