Evaluate the following integrals:
`int sqrt(2ax-x^2dx`

Answer

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

### Step 1: Rewrite the integrand
First, we rewrite the integrand to make it easier to work with. We can factor out a negative sign:
\[ I = \int \sqrt{-(x^2 - 2ax)} \, dx \]
Now, we will complete the square for the expression inside the square root.

### Step 2: Complete the square
The expression \( x^2 - 2ax \) can be rewritten by completing the square:
\[ x^2 - 2ax = (x - a)^2 - a^2 \]
Thus, we have:
\[ I = \int \sqrt{-( (x - a)^2 - a^2 )} \, dx \]
This simplifies to:
\[ I = \int \sqrt{a^2 - (x - a)^2} \, dx \]

### Step 3: Use the standard integral formula
We can now use the standard integral formula:
\[ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \left( \frac{x}{a} \right) + C \]
In our case, we have \( a \) as \( a \) and \( x \) as \( (x - a) \). Therefore, we need to adjust our formula accordingly.

### Step 4: Apply the formula
Using the formula, we substitute \( x \) with \( (x - a) \):
\[ I = \frac{(x - a)}{2} \sqrt{a^2 - (x - a)^2} + \frac{a^2}{2} \sin^{-1} \left( \frac{x - a}{a} \right) + C \]

### Step 5: Simplify the expression
Now, we can simplify the expression:
\[ I = \frac{(x - a)}{2} \sqrt{a^2 - (x - a)^2} + \frac{a^2}{2} \sin^{-1} \left( \frac{x - a}{a} \right) + C \]

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

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