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 can follow these steps:

### Step 1: Substitute \( x = a \cos \theta \)
We start by substituting \( x \) with \( a \cos \theta \). This substitution is useful because it simplifies the expression under the square root.

\[ dx = -a \sin \theta \, d\theta \]

### Step 2: Rewrite the integral
Substituting \( x = a \cos \theta \) and \( dx = -a \sin \theta \, d\theta \) into the integral gives:

\[ I = \int \sqrt{\frac{a + a \cos \theta}{a - a \cos \theta}} (-a \sin \theta) \, d\theta \]

This simplifies to:

\[ I = -a \int \sqrt{\frac{a(1 + \cos \theta)}{a(1 - \cos \theta)}} \sin \theta \, d\theta \]

### Step 3: Simplify the square root
The expression inside the square root can be simplified:

\[ \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} = \sqrt{\frac{2 \cos^2(\theta/2)}{2 \sin^2(\theta/2)}} = \frac{\cos(\theta/2)}{\sin(\theta/2)} \]

### Step 4: Substitute back into the integral
Now substituting this back into the integral gives:

\[ I = -a \int \frac{\cos(\theta/2)}{\sin(\theta/2)} \sin \theta \, d\theta \]

### Step 5: Use half-angle identities
Using the identity \( \sin \theta = 2 \sin(\theta/2) \cos(\theta/2) \):

\[ I = -a \int 2 \cos^2(\theta/2) \, d\theta \]

### Step 6: Integrate
The integral can be computed using the identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \):

\[ I = -a \int 2 \left(\frac{1 + \cos \theta}{2}\right) d\theta = -a \int (1 + \cos \theta) \, d\theta \]

This results in:

\[ I = -a \left( \theta + \sin \theta \right) + C \]

### Step 7: Substitute back for \( \theta \)
Recall that \( \theta = \cos^{-1}\left(\frac{x}{a}\right) \). Thus, we need to express \( \sin \theta \) in terms of \( x \):

\[ \sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{x}{a}\right)^2} = \frac{\sqrt{a^2 - x^2}}{a} \]

### Final Result
Substituting back gives:

\[ I = -a \left( \cos^{-1}\left(\frac{x}{a}\right) + \frac{\sqrt{a^2 - x^2}}{a} \right) + C \]

This simplifies to:

\[ I = -a \cos^{-1}\left(\frac{x}{a}\right) - \sqrt{a^2 - x^2} + C \]