`int_(0)^(2) (dx)/(4+x-x^(2))= `

Options

A. `1/(2sqrt(17))log ((5+sqrt(17))/(5-sqrt(17)))`
B. `1/(2sqrt(17))log ((5-sqrt(17))/(5+sqrt(17)))`
C. `1/(sqrt(17))log ((5+sqrt(17))/(5-sqrt(17)))`
D. `1/(sqrt(17))log ((5-sqrt(17))/(5+sqrt(17)))`

Answer

To solve the integral

\[ I = \int_{0}^{2} \frac{dx}{4 + x - x^2}, \]

we can start by rewriting the denominator.

### Step 1: Rewrite the Denominator
We can rearrange the quadratic expression in the denominator:

\[ 4 + x - x^2 = - (x^2 - x - 4). \]

Thus, we have:

\[ I = - \int_{0}^{2} \frac{dx}{x^2 - x - 4}. \]

### Step 2: Completing the Square
Next, we complete the square for the quadratic \(x^2 - x - 4\):

\[ x^2 - x - 4 = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} - 4 = \left(x - \frac{1}{2}\right)^2 - \frac{17}{4}. \]

So, we can rewrite the integral as:

\[ I = - \int_{0}^{2} \frac{dx}{\left(x - \frac{1}{2}\right)^2 - \frac{17}{4}}. \]

### Step 3: Use the Integral Formula
The form of the integral now resembles the standard integral:

\[ \int \frac{dx}{a^2 - u^2} = \frac{1}{2a} \ln \left| \frac{a + u}{a - u} \right| + C, \]

where \(a = \frac{\sqrt{17}}{2}\) and \(u = x - \frac{1}{2}\).

### Step 4: Set Up the Integral
Thus, we can express the integral as:

\[ I = -\frac{1}{\sqrt{17}} \ln \left| \frac{\frac{\sqrt{17}}{2} + \left(x - \frac{1}{2}\right)}{\frac{\sqrt{17}}{2} - \left(x - \frac{1}{2}\right)} \right| \bigg|_0^2. \]

### Step 5: Evaluate the Limits
Now we evaluate the limits:

1. For \(x = 2\):

\[ I_2 = -\frac{1}{\sqrt{17}} \ln \left| \frac{\frac{\sqrt{17}}{2} + \left(2 - \frac{1}{2}\right)}{\frac{\sqrt{17}}{2} - \left(2 - \frac{1}{2}\right)} \right| = -\frac{1}{\sqrt{17}} \ln \left| \frac{\frac{\sqrt{17}}{2} + \frac{3}{2}}{\frac{\sqrt{17}}{2} - \frac{3}{2}} \right|. \]

2. For \(x = 0\):

\[ I_0 = -\frac{1}{\sqrt{17}} \ln \left| \frac{\frac{\sqrt{17}}{2} + \left(0 - \frac{1}{2}\right)}{\frac{\sqrt{17}}{2} - \left(0 - \frac{1}{2}\right)} \right| = -\frac{1}{\sqrt{17}} \ln \left| \frac{\frac{\sqrt{17}}{2} - \frac{1}{2}}{\frac{\sqrt{17}}{2} + \frac{1}{2}} \right|. \]

### Step 6: Combine the Results
Now we combine the results of the limits:

\[ I = -\frac{1}{\sqrt{17}} \left( \ln \left| \frac{\frac{\sqrt{17}}{2} + \frac{3}{2}}{\frac{\sqrt{17}}{2} - \frac{3}{2}} \right| - \ln \left| \frac{\frac{\sqrt{17}}{2} - \frac{1}{2}}{\frac{\sqrt{17}}{2} + \frac{1}{2}} \right| \right). \]

Using properties of logarithms, we can simplify this to:

\[ I = -\frac{1}{\sqrt{17}} \ln \left( \frac{\frac{\sqrt{17}}{2} + \frac{3}{2}}{\frac{\sqrt{17}}{2} - \frac{3}{2}} \cdot \frac{\frac{\sqrt{17}}{2} + \frac{1}{2}}{\frac{\sqrt{17}}{2} - \frac{1}{2}} \right). \]

### Final Result
After simplifying, we arrive at the final result:

\[ I = \frac{1}{\sqrt{17}} \ln \left( \frac{5 + \sqrt{17}}{5 - \sqrt{17}} \right). \]