Evaluate: `intsqrt(2a x-x^2)\ dx`

Explanation

To evaluate the integral \(\int \sqrt{2ax - x^2} \, dx\), we can follow these steps:

### Step 1: Rewrite the integrand
We start by rewriting the expression under the square root:
\[ \sqrt{2ax - x^2} = \sqrt{-(x^2 - 2ax)} = \sqrt{-(x^2 - 2ax + a^2 - a^2)} = \sqrt{a^2 - (x - a)^2} \]
This transformation allows us to express the integrand in a more manageable form.

### Step 2: Set up the integral
Now we can rewrite the integral:
\[ I = \int \sqrt{a^2 - (x - a)^2} \, dx \]

### Step 3: Use substitution
Let \(t = x - a\). Then, \(dx = dt\) and the limits of integration remain the same since we are not changing the bounds. The integral now becomes:
\[ I = \int \sqrt{a^2 - t^2} \, dt \]

### Step 4: Apply the integral formula
The integral \(\int \sqrt{a^2 - t^2} \, dt\) can be evaluated using the formula:
\[ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \left(\frac{x}{a}\right) + C \]
Applying this formula to our integral:
\[ I = \frac{t}{2} \sqrt{a^2 - t^2} + \frac{a^2}{2} \sin^{-1} \left(\frac{t}{a}\right) + C \]

### Step 5: Substitute back
Now, we substitute back \(t = x - a\):
\[ I = \frac{x - a}{2} \sqrt{a^2 - (x - a)^2} + \frac{a^2}{2} \sin^{-1} \left(\frac{x - a}{a}\right) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ I = \frac{x - a}{2} \sqrt{a^2 - (x - a)^2} + \frac{a^2}{2} \sin^{-1} \left(\frac{x - a}{a}\right) + C \]