Evaluate `int_(0)^(1)(1)/(3+4x)d x`

Answer

To evaluate the integral \( \int_{0}^{1} \frac{1}{3 + 4x} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = 3 + 4x \). Then, we differentiate both sides to find \( dx \):
\[ dt = 4 \, dx \quad \Rightarrow \quad dx = \frac{1}{4} dt \]

### Step 2: Change the limits of integration
Now we need to change the limits of integration according to our substitution:
- When \( x = 0 \):
\[ t = 3 + 4(0) = 3 \]
- When \( x = 1 \):
\[ t = 3 + 4(1) = 7 \]
Thus, the new limits of integration are from \( t = 3 \) to \( t = 7 \).

### Step 3: Substitute in the integral
Now we can substitute \( t \) and \( dx \) into the integral:
\[ \int_{0}^{1} \frac{1}{3 + 4x} \, dx = \int_{3}^{7} \frac{1}{t} \cdot \frac{1}{4} \, dt \]
This simplifies to:
\[ \frac{1}{4} \int_{3}^{7} \frac{1}{t} \, dt \]

### Step 4: Integrate
The integral of \( \frac{1}{t} \) is \( \ln |t| \):
\[ \frac{1}{4} \left[ \ln |t| \right]_{3}^{7} = \frac{1}{4} \left( \ln 7 - \ln 3 \right) \]

### Step 5: Simplify using logarithmic properties
Using the property of logarithms \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \):
\[ \frac{1}{4} \left( \ln 7 - \ln 3 \right) = \frac{1}{4} \ln \left( \frac{7}{3} \right) \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{0}^{1} \frac{1}{3 + 4x} \, dx = \frac{1}{4} \ln \left( \frac{7}{3} \right) \]