If `inte^(((x^(6)+x^(2)-x-1)/(x-1)))(5x^(4)+4x^(3)+3x^(2)+2x+2)dx=e^(g(x))+C` where C is constant of integration and g(0) = 1, then the value of g (1) is___________ .

Answer

To solve the given problem, we need to evaluate the integral and find the function \( g(1) \) based on the information provided. Let's break down the solution step by step.

### Step 1: Set Up the Integral
We start with the expression given in the problem:

\[ \int e^{\frac{x^6 + x^2 - x - 1}{x - 1}} (5x^4 + 4x^3 + 3x^2 + 2x + 2) \, dx = e^{g(x)} + C \]

### Step 2: Simplify the Exponent
The exponent can be simplified as follows:

\[ \frac{x^6 + x^2 - x - 1}{x - 1} \]

Using polynomial long division, we can divide \( x^6 + x^2 - x - 1 \) by \( x - 1 \). After performing the division, we find that:

\[ x^6 + x^2 - x - 1 = (x - 1)(x^5 + x^4 + x^3 + x^2 + x + 1) + 0 \]

Thus, the expression simplifies to:

\[ \int e^{x^5 + x^4 + x^3 + x^2 + x + 1} (5x^4 + 4x^3 + 3x^2 + 2x + 2) \, dx \]

### Step 3: Recognize the Integral
Now, we can recognize that the integral can be expressed in terms of \( e^{g(x)} \):

\[ g(x) = x^5 + x^4 + x^3 + x^2 + x + 1 \]

### Step 4: Find \( g(0) \)
We are given that \( g(0) = 1 \). Let's calculate \( g(0) \):

\[ g(0) = 0^5 + 0^4 + 0^3 + 0^2 + 0 + 1 = 1 \]

This confirms the condition given in the problem.

### Step 5: Find \( g(1) \)
Now, we need to find \( g(1) \):

\[ g(1) = 1^5 + 1^4 + 1^3 + 1^2 + 1 + 1 \]

Calculating this gives:

\[ g(1) = 1 + 1 + 1 + 1 + 1 + 1 = 6 \]

### Final Answer
Thus, the value of \( g(1) \) is:

\[ \boxed{6} \]