Evaluate the following integrals
`int sqrt((a-x)/(x-b))dx`

Answer

To evaluate the integral \( I = \int \sqrt{\frac{a-x}{x-b}} \, dx \), we will follow a systematic approach using a trigonometric substitution.

### Step-by-Step Solution:

1. **Substitution**: Let \( x = a \cos^2 \theta + b \sin^2 \theta \). This substitution helps to simplify the integral.
- **Hint**: This substitution is chosen to express \( a-x \) and \( x-b \) in terms of trigonometric functions.

2. **Differentiate**: Compute \( dx \):
\[ dx = \frac{d}{d\theta}(a \cos^2 \theta + b \sin^2 \theta) = -2a \cos \theta \sin \theta + 2b \sin \theta \cos \theta = (2b - 2a) \sin \theta \cos \theta \, d\theta = (b-a) \sin 2\theta \, d\theta. \]
- **Hint**: Use the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \) to simplify the expression.

3. **Substituting in the Integral**:
Substitute \( x \) and \( dx \) into the integral:
\[ I = \int \sqrt{\frac{a - (a \cos^2 \theta + b \sin^2 \theta)}{(a \cos^2 \theta + b \sin^2 \theta) - b}} \cdot (b-a) \sin 2\theta \, d\theta. \]
Simplifying the expression inside the square root:
\[ a - (a \cos^2 \theta + b \sin^2 \theta) = a(1 - \cos^2 \theta) - b \sin^2 \theta = a \sin^2 \theta - b \sin^2 \theta = (a-b) \sin^2 \theta. \]
And for the denominator:
\[ (a \cos^2 \theta + b \sin^2 \theta) - b = a \cos^2 \theta - b(1 - \sin^2 \theta) = a \cos^2 \theta - b + b \sin^2 \theta = a \cos^2 \theta + (b-a) \sin^2 \theta. \]
- **Hint**: Rearranging terms helps to simplify the integral.

4. **Final Form of the Integral**:
Now, the integral becomes:
\[ I = (b-a) \int \sqrt{\frac{(a-b) \sin^2 \theta}{a \cos^2 \theta + (b-a) \sin^2 \theta}} \cdot \sin 2\theta \, d\theta. \]
- **Hint**: Factor out constants and simplify the integral further.

5. **Simplifying Further**:
The integral can be simplified using trigonometric identities. We can express \( \sin^2 \theta \) and \( \cos^2 \theta \) in terms of \( \tan \theta \) to facilitate integration.
- **Hint**: Consider using \( \tan \theta \) substitution to express everything in terms of \( \tan \theta \).

6. **Integration**:
After simplification, we can integrate using standard integral formulas. The integral of \( \sin^2 \theta \) can be expressed in terms of \( \theta \) and \( \cos 2\theta \).
- **Hint**: Use the identity \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \).

7. **Back Substitution**:
Once the integral is evaluated, substitute back \( \theta \) in terms of \( x \). Recall that:
\[ \tan \theta = \sqrt{\frac{a-x}{x-b}}. \]
- **Hint**: Use the inverse tangent function to revert back to \( x \).

8. **Final Result**:
The final result will be expressed in terms of \( x \) and will include a constant of integration \( C \).

### Final Answer:
\[ I = (b-a) \left( \text{expression involving } x + C \right). \]