Let `f(x)=int_(0)^x( (a-1)(t^2+t+1)^2-(a+1)(t^4+t^2+1))dt` then find the total number of integral value of `a` for which `f'(x)=0` has no real root

Answer

To solve the problem, we need to find the total number of integral values of \( a \) for which the equation \( f'(x) = 0 \) has no real roots.

### Step 1: Find \( f'(x) \)

Given:
\[ f(x) = \int_{0}^{x} \left( (a-1)(t^2 + t + 1)^2 - (a+1)(t^4 + t^2 + 1) \right) dt \]

Using the Fundamental Theorem of Calculus, we differentiate \( f(x) \):
\[ f'(x) = (a-1)(x^2 + x + 1)^2 - (a+1)(x^4 + x^2 + 1) \]

### Step 2: Set \( f'(x) = 0 \)

We need to solve:
\[ (a-1)(x^2 + x + 1)^2 - (a+1)(x^4 + x^2 + 1) = 0 \]

### Step 3: Rearranging the equation

Rearranging gives:
\[ (a-1)(x^2 + x + 1)^2 = (a+1)(x^4 + x^2 + 1) \]

### Step 4: Analyze the quadratic form

This is a quadratic equation in \( x \). For the quadratic equation \( Ax^2 + Bx + C = 0 \) to have no real roots, the discriminant must be less than zero:
\[ D = B^2 - 4AC < 0 \]

### Step 5: Identify coefficients

From our equation, we can identify:
- \( A = a + 1 \)
- \( B = -a \)
- \( C = (a-1) - (a+1) = -2 \)

### Step 6: Calculate the discriminant

The discriminant \( D \) is:
\[ D = (-a)^2 - 4(a + 1)(-2) = a^2 + 8(a + 1) \]
\[ D = a^2 + 8a + 8 \]

### Step 7: Set the discriminant less than zero

We need:
\[ a^2 + 8a + 8 < 0 \]

### Step 8: Find the roots of the quadratic

To find the roots of \( a^2 + 8a + 8 = 0 \), we use the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-8 \pm \sqrt{64 - 32}}{2} = \frac{-8 \pm \sqrt{32}}{2} = \frac{-8 \pm 4\sqrt{2}}{2} = -4 \pm 2\sqrt{2} \]

### Step 9: Determine the intervals

The roots are:
\[ a_1 = -4 - 2\sqrt{2}, \quad a_2 = -4 + 2\sqrt{2} \]

### Step 10: Analyze the intervals

The quadratic opens upwards (since the coefficient of \( a^2 \) is positive). Thus, \( a^2 + 8a + 8 < 0 \) between the roots:
\[ -4 - 2\sqrt{2} < a < -4 + 2\sqrt{2} \]

### Step 11: Find integral values of \( a \)

Calculating the approximate values:
- \( -4 - 2\sqrt{2} \approx -4 - 2(1.414) \approx -6.828 \)
- \( -4 + 2\sqrt{2} \approx -4 + 2(1.414) \approx -1.172 \)

Thus, the integral values of \( a \) in the interval \( (-6.828, -1.172) \) are:
- \( -6, -5, -4, -3, -2 \)

### Conclusion

The total number of integral values of \( a \) for which \( f'(x) = 0 \) has no real roots is:
\[ \boxed{5} \]