`int_(2)^(5)(1)/((2+3x))dx` is -

Options

A. `(15)/(2)`
B. `(7)/(5)`
C. `(1)/(3)l"n"(17)/(8)`
D. `l"n"(17)/(8)`

Explanation

To solve the integral \( \int_{2}^{5} \frac{1}{2 + 3x} \, dx \), we will follow these steps:

### Step 1: Substitution
Let \( t = 2 + 3x \). Then, differentiate both sides with respect to \( x \):
\[ dt = 3 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3} \]

### Step 2: Change the limits of integration
When \( x = 2 \):
\[ t = 2 + 3(2) = 2 + 6 = 8 \]
When \( x = 5 \):
\[ t = 2 + 3(5) = 2 + 15 = 17 \]
Thus, the limits of integration change from \( x = 2 \) to \( x = 5 \) into \( t = 8 \) to \( t = 17 \).

### Step 3: Substitute into the integral
Now substitute \( t \) and \( dx \) into the integral:
\[ \int_{2}^{5} \frac{1}{2 + 3x} \, dx = \int_{8}^{17} \frac{1}{t} \cdot \frac{dt}{3} \]
This simplifies to:
\[ \frac{1}{3} \int_{8}^{17} \frac{1}{t} \, dt \]

### Step 4: Integrate
The integral of \( \frac{1}{t} \) is \( \ln |t| \):
\[ \frac{1}{3} \left[ \ln |t| \right]_{8}^{17} \]
Calculating this gives:
\[ \frac{1}{3} \left( \ln 17 - \ln 8 \right) = \frac{1}{3} \ln \left( \frac{17}{8} \right) \]

### Step 5: Final answer
Thus, the value of the integral is:
\[ \frac{1}{3} \ln \left( \frac{17}{8} \right) \]