If the two `AB:(int_(0)^(2t)((sinx)/x+1)dx)x+y=3t` and `AC:2tx+y=0` intersect at a point A the x-coordinate of a point A as `t to 0`, is equal to `p/q` (p and q are in their lowest form) the `(p+q)` is ………….

Answer

To solve the problem step-by-step, we need to find the x-coordinate of the intersection point \( A \) of the two lines \( AB \) and \( AC \) as \( t \) approaches 0.

### Step 1: Define the equations of the lines
The equations of the lines are given as:
1. \( AB: \int_{0}^{2t} \left(\frac{\sin x}{x} + 1\right) dx \cdot x + y = 3t \)
2. \( AC: 2tx + y = 0 \)

### Step 2: Express \( y \) in terms of \( x \) for line \( AC \)
From the equation of line \( AC \):
\[ y = -2tx \]

### Step 3: Substitute \( y \) into the equation of line \( AB \)
Substituting \( y = -2tx \) into the equation of line \( AB \):
\[ \int_{0}^{2t} \left(\frac{\sin x}{x} + 1\right) dx \cdot x - 2tx = 3t \]
This simplifies to:
\[ \int_{0}^{2t} \left(\frac{\sin x}{x} + 1\right) dx \cdot x = 3t + 2tx \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ \int_{0}^{2t} \left(\frac{\sin x}{x} + 1\right) dx \cdot x - 2tx - 3t = 0 \]

### Step 5: Analyze the limit as \( t \to 0 \)
We need to find the limit of \( x \) as \( t \) approaches 0. First, we evaluate the integral:
\[ \int_{0}^{2t} \left(\frac{\sin x}{x} + 1\right) dx \]
As \( t \to 0 \), the integral approaches:
\[ \int_{0}^{0} \left(\frac{\sin x}{x} + 1\right) dx = 0 \]

### Step 6: Apply L'Hôpital's Rule
Since we have a \( 0/0 \) form, we apply L'Hôpital's Rule. We differentiate the numerator and denominator with respect to \( t \):
- The numerator becomes \( 3 \).
- The denominator involves differentiating the integral:
\[ \frac{d}{dt} \left( \int_{0}^{2t} \left(\frac{\sin x}{x} + 1\right) dx \cdot x \right) \]
Using the Leibniz rule, we differentiate:
\[ = \left(\frac{\sin(2t)}{2t} + 1\right) \cdot 2t - \int_{0}^{2t} \left(\frac{\sin x}{x} + 1\right) dx \cdot 2 \]

### Step 7: Evaluate the limit
As \( t \to 0 \):
- \( \frac{\sin(2t)}{2t} \to 1 \)
- The integral simplifies to \( 0 \).

Thus, we have:
\[ x = \frac{3}{2} \text{ as } t \to 0 \]

### Step 8: Express \( x \) in the form \( \frac{p}{q} \)
Here, \( x = \frac{3}{2} \) implies \( p = 3 \) and \( q = 2 \).

### Step 9: Find \( p + q \)
Finally, we compute:
\[ p + q = 3 + 2 = 5 \]

### Final Answer
The value of \( p + q \) is \( \boxed{5} \).