Evaluate: `int(8x-11)/sqrt(5+2x-x^(2))`dx

Answer

To evaluate the integral

\[ I = \int \frac{8x - 11}{\sqrt{5 + 2x - x^2}} \, dx, \]

we can break it down into manageable parts.

### Step 1: Rewrite the integrand
We start by rewriting the numerator:

\[ 8x - 11 = 8(x - 1) - 3. \]

This allows us to express the integral as:

\[ I = \int \frac{8(x - 1) - 3}{\sqrt{5 + 2x - x^2}} \, dx = \int \frac{8(x - 1)}{\sqrt{5 + 2x - x^2}} \, dx - \int \frac{3}{\sqrt{5 + 2x - x^2}} \, dx. \]

Let’s denote these two integrals as \( I_1 \) and \( I_2 \):

\[ I_1 = \int \frac{8(x - 1)}{\sqrt{5 + 2x - x^2}} \, dx, \]
\[ I_2 = \int \frac{3}{\sqrt{5 + 2x - x^2}} \, dx. \]

### Step 2: Simplify \( I_1 \)
For \( I_1 \):

\[ I_1 = 8 \int \frac{x - 1}{\sqrt{5 + 2x - x^2}} \, dx. \]

Now, we can use a substitution. Let:

\[ u = 5 + 2x - x^2. \]

Then, we differentiate \( u \):

\[ \frac{du}{dx} = 2 - 2x \implies du = (2 - 2x) \, dx \implies dx = \frac{du}{2 - 2x} = \frac{du}{2(1 - x)}. \]

Now, we can express \( x - 1 \) in terms of \( u \):

From \( u = 5 + 2x - x^2 \), we can rearrange to find \( x \):

\[ x^2 - 2x + (u - 5) = 0. \]

Using the quadratic formula:

\[ x = \frac{2 \pm \sqrt{4 - 4(u - 5)}}{2} = 1 \pm \sqrt{9 - u}. \]

We take \( x - 1 = \sqrt{9 - u} \) (considering the positive root for simplicity).

Substituting back into \( I_1 \):

\[ I_1 = 8 \int \frac{\sqrt{9 - u}}{\sqrt{u}} \cdot \frac{du}{2(1 - (1 + \sqrt{9 - u}))} = 4 \int \frac{\sqrt{9 - u}}{\sqrt{u}} \, du. \]

### Step 3: Solve \( I_2 \)
For \( I_2 \):

\[ I_2 = 3 \int \frac{1}{\sqrt{5 + 2x - x^2}} \, dx. \]

Using the same substitution \( u = 5 + 2x - x^2 \):

\[ I_2 = 3 \int \frac{1}{\sqrt{u}} \cdot \frac{du}{2 - 2x} = \frac{3}{2} \int \frac{du}{\sqrt{u}}. \]

### Step 4: Integrate both parts
Now we can integrate both parts:

1. For \( I_1 \):

\[ I_1 = 4 \int \frac{\sqrt{9 - u}}{\sqrt{u}} \, du. \]

This integral can be solved using trigonometric substitution or other methods.

2. For \( I_2 \):

\[ I_2 = \frac{3}{2} \cdot 2\sqrt{u} = 3\sqrt{u}. \]

### Step 5: Combine results
Finally, we combine the results of \( I_1 \) and \( I_2 \) and substitute back for \( u \):

\[ I = I_1 + I_2 = 4 \int \frac{\sqrt{9 - (5 + 2x - x^2)}}{\sqrt{5 + 2x - x^2}} \, dx + 3\sqrt{5 + 2x - x^2} + C. \]

### Final Answer
The final answer will be:

\[ I = 4 \int \frac{\sqrt{4 - (2x - 1)^2}}{\sqrt{5 + 2x - x^2}} \, dx + 3\sqrt{5 + 2x - x^2} + C. \]