Let `f (x) =x ^(2) +2x -t ^(2) and f(x)=0` has two root `alpha (t ) and beta (t) (alpha lt beta)` where t is a real parameter. Let `I (t)=int _(alpha ) ^(beta)f (x)` dx. If the maximum value of `I (t)` be `lamda and | lamda| =p/q` where p and q are relatively prime positive integers. Find the product (pq).

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and break it down into clear steps.

### Step 1: Define the Function and Find the Roots
Given the function:
\[ f(x) = x^2 + 2x - t^2 \]

To find the roots \( \alpha(t) \) and \( \beta(t) \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 2 \), and \( c = -t^2 \).

Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4(1)(-t^2) = 4 + 4t^2 = 4(1 + t^2) \]

Now applying the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{4(1 + t^2)}}{2} = -1 \pm \sqrt{1 + t^2} \]

Thus, the roots are:
\[ \alpha(t) = -1 - \sqrt{1 + t^2} \]
\[ \beta(t) = -1 + \sqrt{1 + t^2} \]

### Step 2: Set Up the Integral
We need to evaluate the integral:
\[ I(t) = \int_{\alpha(t)}^{\beta(t)} f(x) \, dx \]

### Step 3: Evaluate the Integral
First, we compute the integral of \( f(x) \):
\[ \int f(x) \, dx = \int (x^2 + 2x - t^2) \, dx = \frac{x^3}{3} + x^2 - t^2 x + C \]

Now we evaluate the definite integral from \( \alpha(t) \) to \( \beta(t) \):
\[ I(t) = \left[ \frac{x^3}{3} + x^2 - t^2 x \right]_{\alpha(t)}^{\beta(t)} \]

Calculating \( I(t) \):
\[ I(t) = \left( \frac{\beta(t)^3}{3} + \beta(t)^2 - t^2 \beta(t) \right) - \left( \frac{\alpha(t)^3}{3} + \alpha(t)^2 - t^2 \alpha(t) \right) \]

### Step 4: Simplifying the Expression
Using the roots:
\[ \beta(t) = -1 + \sqrt{1 + t^2}, \quad \alpha(t) = -1 - \sqrt{1 + t^2} \]

We can substitute these values into the expression for \( I(t) \) and simplify.

### Step 5: Finding the Maximum Value
To find the maximum value of \( I(t) \), we differentiate \( I(t) \) with respect to \( t \) and set the derivative to zero:
\[ I'(t) = 0 \]

After finding the critical points, we can evaluate \( I(t) \) at these points to find the maximum value \( \lambda \).

### Step 6: Finding \( | \lambda | \)
From the calculations, we find that the maximum value \( \lambda \) is:
\[ \lambda = -\frac{4}{3} \]

Thus, we have:
\[ | \lambda | = \frac{4}{3} \]

### Step 7: Express \( | \lambda | \) as \( \frac{p}{q} \)
Here, \( p = 4 \) and \( q = 3 \) are relatively prime integers.

### Step 8: Calculate the Product \( pq \)
Finally, we calculate:
\[ pq = 4 \times 3 = 12 \]

### Final Answer
The product \( pq \) is:
\[ \boxed{12} \]