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

Answer

To evaluate the integral

\[ I = \int_{0}^{1} \frac{1}{3 + 4x} \, dx, \]

we will follow these steps:

### Step 1: Substitution
Let \( v = 3 + 4x \). Then, differentiate both sides to find \( dx \):

\[ dv = 4 \, dx \quad \Rightarrow \quad dx = \frac{1}{4} \, dv. \]

### Step 2: Change the limits of integration
Now we need to change the limits of integration according to our substitution.

- When \( x = 0 \):
\[ v = 3 + 4(0) = 3. \]

- When \( x = 1 \):
\[ v = 3 + 4(1) = 7. \]

Thus, the new limits of integration are from \( v = 3 \) to \( v = 7 \).

### Step 3: Rewrite the integral
Substituting \( v \) and \( dx \) into the integral, we get:

\[ I = \int_{3}^{7} \frac{1}{v} \cdot \frac{1}{4} \, dv = \frac{1}{4} \int_{3}^{7} \frac{1}{v} \, dv. \]

### Step 4: Evaluate the integral
The integral of \( \frac{1}{v} \) is \( \log v \). Therefore, we have:

\[ I = \frac{1}{4} \left[ \log v \right]_{3}^{7} = \frac{1}{4} \left( \log 7 - \log 3 \right). \]

### Step 5: Simplify using properties of logarithms
Using the property \( \log a - \log b = \log \left( \frac{a}{b} \right) \), we can simplify:

\[ I = \frac{1}{4} \log \left( \frac{7}{3} \right). \]

### Final Answer
Thus, the value of the integral is:

\[ I = \frac{1}{4} \log \left( \frac{7}{3} \right). \]

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