`int_0^1 3^x dx`

Answer

To solve the integral \( \int_0^1 3^x \, dx \), we will follow these steps:

### Step 1: Identify the integral
We need to evaluate the integral:
\[ I = \int_0^1 3^x \, dx \]

### Step 2: Use the formula for the integral of an exponential function
The integral of \( a^x \) is given by the formula:
\[ \int a^x \, dx = \frac{a^x}{\ln a} + C \]
where \( C \) is the constant of integration and \( a > 0 \), \( a \neq 1 \).

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

### Step 4: Evaluate the definite integral from 0 to 1
Now we need to evaluate this from 0 to 1:
\[ I = \left[ \frac{3^x}{\ln 3} \right]_0^1 \]
This means we will substitute the upper limit (1) and the lower limit (0) into the expression.

### Step 5: Substitute the upper limit
Substituting \( x = 1 \):
\[ \frac{3^1}{\ln 3} = \frac{3}{\ln 3} \]

### Step 6: Substitute the lower limit
Substituting \( x = 0 \):
\[ \frac{3^0}{\ln 3} = \frac{1}{\ln 3} \]

### Step 7: Calculate the definite integral
Now, we subtract the lower limit from the upper limit:
\[ I = \frac{3}{\ln 3} - \frac{1}{\ln 3} = \frac{3 - 1}{\ln 3} = \frac{2}{\ln 3} \]

### Final Result
Thus, the value of the integral is:
\[ \int_0^1 3^x \, dx = \frac{2}{\ln 3} \]

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