`int(sqrt((a+x)/(a-x))+sqrt((a-x)/(a+x)))dx " is equal to "`

Options

A. `2sin^(-1)(x//a)+c`
B. `2asin^(-1)(x//a)+c`
C. `2cos^(-1)(x//a)+c`
D. `2acos^(-1)(x//a)+c`

Explanation

To solve the integral \( \int \left( \sqrt{\frac{a+x}{a-x}} + \sqrt{\frac{a-x}{a+x}} \right) dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \left( \sqrt{\frac{a+x}{a-x}} + \sqrt{\frac{a-x}{a+x}} \right) dx \]

### Step 2: Combine the Terms
We can combine the two square root terms under a common denominator:
\[ I = \int \left( \frac{\sqrt{(a+x)(a+x)} + \sqrt{(a-x)(a-x)}}{\sqrt{(a-x)(a+x)}} \right) dx \]
This simplifies to:
\[ I = \int \left( \frac{\sqrt{(a+x)^2} + \sqrt{(a-x)^2}}{\sqrt{(a-x)(a+x)}} \right) dx \]
\[ = \int \left( \frac{(a+x) + (a-x)}{\sqrt{(a-x)(a+x)}} \right) dx \]
\[ = \int \left( \frac{2a}{\sqrt{(a-x)(a+x)}} \right) dx \]

### Step 3: Simplify the Integral
Now we can rewrite the integral as:
\[ I = 2a \int \frac{1}{\sqrt{a^2 - x^2}} dx \]

### Step 4: Solve the Integral
The integral \( \int \frac{1}{\sqrt{a^2 - x^2}} dx \) is a standard integral that evaluates to:
\[ \int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C \]
Thus, we have:
\[ I = 2a \sin^{-1}\left(\frac{x}{a}\right) + C \]

### Final Answer
The final result for the integral is:
\[ \int \left( \sqrt{\frac{a+x}{a-x}} + \sqrt{\frac{a-x}{a+x}} \right) dx = 2a \sin^{-1}\left(\frac{x}{a}\right) + C \]