Let `f:R rarr R` be a function as
`f(x)=(x-1)(x+2)(x-3)(x-6)-100`. If `g(x)` is a polynomial of degree `le 3` such that `int (g(x))/(f(x))dx` does not contain any logarithm function and `g(-2)=10`. Then
`int (g(x))/(f(x))dx`, equals

Options

A. `tan^(-1)((x-2)/(2))+c`
B. `tan^(-1)((x-1)/(1))+c`
C. `tan^(-1)(x)+c`
D. None of these

Answer

To solve the problem step by step, we need to find the integral of \( \frac{g(x)}{f(x)} \, dx \) where \( f(x) = (x-1)(x+2)(x-3)(x-6) - 100 \) and \( g(-2) = 10 \). We also need to ensure that the integral does not contain any logarithmic functions.

### Step 1: Expand \( f(x) \)

First, we need to expand the function \( f(x) \).

\[ f(x) = (x-1)(x+2)(x-3)(x-6) - 100 \]

Calculating \( (x-1)(x+2) \):

\[ (x-1)(x+2) = x^2 + x - 2 \]

Calculating \( (x-3)(x-6) \):

\[ (x-3)(x-6) = x^2 - 9x + 18 \]

Now, we multiply these two results:

\[ f(x) = (x^2 + x - 2)(x^2 - 9x + 18) - 100 \]

Expanding this product:

\[ = x^4 - 9x^3 + 18x^2 + x^3 - 9x^2 - 18 - 2x^2 + 18x - 36 - 100 \]

Combining like terms:

\[ = x^4 - 8x^3 + 7x^2 + 18x - 154 \]

### Step 2: Set up the integral

We need to find \( \int \frac{g(x)}{f(x)} \, dx \) where \( g(x) \) is a polynomial of degree \( \leq 3 \).

### Step 3: Use Partial Fraction Decomposition

Since \( f(x) \) is a polynomial of degree 4, we can express \( \frac{g(x)}{f(x)} \) using partial fractions. We assume:

\[ \frac{g(x)}{f(x)} = \frac{A}{x^2 - 4x + 8} + \frac{B}{x^2 - 4x - 17} \]

where \( A \) and \( B \) are polynomials of degree less than or equal to 1.

### Step 4: Determine conditions for \( g(x) \)

For the integral to not contain logarithmic functions, we need the linear terms in the numerators of the partial fractions to vanish. Thus, we set:

\[ A = 0 \quad \text{and} \quad C = 0 \]

This means \( g(x) \) must be a constant polynomial.

### Step 5: Find \( g(x) \)

Since \( g(-2) = 10 \), we can conclude:

\[ g(x) = 10 \]

### Step 6: Evaluate the integral

Now we evaluate:

\[ \int \frac{10}{f(x)} \, dx \]

### Step 7: Final result

The integral simplifies to:

\[ \int \frac{10}{(x-1)(x+2)(x-3)(x-6) - 100} \, dx \]

This integral can be evaluated using appropriate techniques, but we have established that \( g(x) = 10 \) and \( f(x) \) is as defined above.

### Conclusion

Thus, the final answer for the integral \( \int \frac{g(x)}{f(x)} \, dx \) is:

\[ \int \frac{10}{f(x)} \, dx \]