Evaluate the following integrals:
`int sqrt(frac{a+x}{a-x}dx`

Answer

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

### Step 1: Trigonometric Substitution
Let us substitute \( x = a \cos(2\theta) \). Then, we differentiate to find \( dx \):
\[ dx = -2a \sin(2\theta) \, d\theta \]

### Step 2: Substitute in the Integral
Now substitute \( x \) and \( dx \) into the integral:
\[ I = \int \sqrt{\frac{a + a \cos(2\theta)}{a - a \cos(2\theta)}} (-2a \sin(2\theta)) \, d\theta \]

### Step 3: Simplify the Expression
Factor out \( a \) in the square root:
\[ I = \int \sqrt{\frac{a(1 + \cos(2\theta))}{a(1 - \cos(2\theta))}} (-2a \sin(2\theta)) \, d\theta \]
This simplifies to:
\[ I = \int \sqrt{\frac{1 + \cos(2\theta)}{1 - \cos(2\theta)}} (-2a \sin(2\theta)) \, d\theta \]

### Step 4: Use Trigonometric Identities
Using the identities \( 1 + \cos(2\theta) = 2 \cos^2(\theta) \) and \( 1 - \cos(2\theta) = 2 \sin^2(\theta) \):
\[ I = \int \sqrt{\frac{2 \cos^2(\theta)}{2 \sin^2(\theta)}} (-2a \sin(2\theta)) \, d\theta \]
This reduces to:
\[ I = \int \frac{\cos(\theta)}{\sin(\theta)} (-2a \cdot 2 \sin(\theta) \cos(\theta)) \, d\theta \]
\[ I = -4a \int \cos^2(\theta) \, d\theta \]

### Step 5: Integrate \( \cos^2(\theta) \)
Using the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \):
\[ I = -4a \int \frac{1 + \cos(2\theta)}{2} \, d\theta \]
\[ I = -2a \left( \theta + \frac{\sin(2\theta)}{2} \right) + C \]

### Step 6: Back Substitute for \( \theta \)
Recall that \( \theta = \frac{1}{2} \cos^{-1}\left(\frac{x}{a}\right) \). Thus:
\[ I = -2a \left( \frac{1}{2} \cos^{-1}\left(\frac{x}{a}\right) + \frac{\sin\left(\cos^{-1}\left(\frac{x}{a}\right)\right)}{2} \right) + C \]
Using the relationship \( \sin\left(\cos^{-1}\left(\frac{x}{a}\right)\right) = \sqrt{1 - \left(\frac{x}{a}\right)^2} = \frac{\sqrt{a^2 - x^2}}{a} \):
\[ I = -a \cos^{-1}\left(\frac{x}{a}\right) - \sqrt{a^2 - x^2} + C \]

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