`int_2^3 3^x dx=`___

Answer

To solve the integral \( \int_2^3 3^x \, dx \), we will follow these steps:

### Step 1: Identify the Integral Formula
We know that the integral of \( a^x \) with respect to \( x \) is given by the formula:
\[ \int a^x \, dx = \frac{a^x}{\ln a} + C \]
where \( a \) is a constant and \( C \) is the constant of integration.

### Step 2: Apply the Formula
In our case, \( a = 3 \). Therefore, we can apply the formula:
\[ \int 3^x \, dx = \frac{3^x}{\ln 3} + C \]

### Step 3: Evaluate the Definite Integral
Now, we need to evaluate the definite integral from 2 to 3:
\[ \int_2^3 3^x \, dx = \left[ \frac{3^x}{\ln 3} \right]_2^3 \]
This means we will substitute the upper limit (3) and the lower limit (2) into the expression.

### Step 4: Substitute the Upper Limit
First, we substitute the upper limit \( x = 3 \):
\[ \frac{3^3}{\ln 3} = \frac{27}{\ln 3} \]

### Step 5: Substitute the Lower Limit
Next, we substitute the lower limit \( x = 2 \):
\[ \frac{3^2}{\ln 3} = \frac{9}{\ln 3} \]

### Step 6: Calculate the Result
Now, we subtract the result of the lower limit from the result of the upper limit:
\[ \int_2^3 3^x \, dx = \frac{27}{\ln 3} - \frac{9}{\ln 3} = \frac{27 - 9}{\ln 3} = \frac{18}{\ln 3} \]

### Final Answer
Thus, the value of the integral \( \int_2^3 3^x \, dx \) is:
\[ \frac{18}{\ln 3} \]

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