If ` int (2x + 5 )/(sqrt(7 - 6 x - x ^ 2 )) dx = A sqrt (7 - 6x - x ^ 2 ) + B sin ^ ( -1) (( x + 3 )/( 4 ) ) + C ` (Where C is a constant of integration), then the ordered pair ` (A, B)` is equal to :

Options

A. ` ( -2, - 1) `
B. ` (2 , -1 ) `
C. ` ( -2, 1 ) `
D. ` (2, 1 ) `

Answer

To solve the integral

\[ I = \int \frac{2x + 5}{\sqrt{7 - 6x - x^2}} \, dx \]

we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:

\[ I = \int \frac{2x + 5}{\sqrt{7 - 6x - x^2}} \, dx \]

To simplify the integration, we can manipulate the numerator. We can express \(2x + 5\) in a way that will help us use the derivative of the denominator.

### Step 2: Differentiate the Denominator
The denominator is \(7 - 6x - x^2\). Let's find its derivative:

\[ \frac{d}{dx}(7 - 6x - x^2) = -6 - 2x \]

This means that:

\[ d(7 - 6x - x^2) = (-6 - 2x) \, dx \]

### Step 3: Manipulate the Numerator
Now, we can rewrite the numerator \(2x + 5\) as follows:

\[ 2x + 5 = -(-2x - 5) = -(-2x - 6 + 1) = -(-2x - 6) + 1 \]

So, we can write:

\[ I = \int \frac{-(-2x - 6) + 1}{\sqrt{7 - 6x - x^2}} \, dx \]

### Step 4: Split the Integral
This allows us to split the integral into two parts:

\[ I = -\int \frac{-2x - 6}{\sqrt{7 - 6x - x^2}} \, dx + \int \frac{1}{\sqrt{7 - 6x - x^2}} \, dx \]

### Step 5: Substitute for \(t\)
Let \(t = 7 - 6x - x^2\). Then, we can express \(dx\) in terms of \(dt\):

\[ dt = (-6 - 2x) \, dx \implies dx = \frac{dt}{-6 - 2x} \]

### Step 6: Change of Variables
Now, we can change the variables in the integrals. The first integral becomes:

\[ -\int \frac{-2x - 6}{\sqrt{t}} \frac{dt}{-6 - 2x} \]

And the second integral can be simplified using the identity for the inverse sine function.

### Step 7: Solve the Integrals
After performing the integration, we find that:

\[ I = -2\sqrt{7 - 6x - x^2} - \sin^{-1}\left(\frac{x + 3}{4}\right) + C \]

### Step 8: Compare with Given Expression
We compare our result with the given expression:

\[ A\sqrt{7 - 6x - x^2} + B\sin^{-1}\left(\frac{x + 3}{4}\right) + C \]

From this comparison, we can identify:

- \(A = -2\)
- \(B = -1\)

### Final Answer
Thus, the ordered pair \((A, B)\) is:

\[ \boxed{(-2, -1)} \]