Evaluate `int x sqrt((a^(2)-x^(2))/(a^(2)+x^(2)))dx`

Answer

To evaluate the integral \( \int x \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} \, dx \), we will use a trigonometric substitution. Let's proceed step by step.

### Step 1: Substitution
Let \( x^2 = a^2 \cos(2\theta) \). Then, differentiating both sides with respect to \( x \):
\[ 2x \, dx = -a^2 \sin(2\theta) \, d\theta \]
This gives us:
\[ x \, dx = -\frac{a^2}{2} \sin(2\theta) \, d\theta \]

### Step 2: Rewrite the Integral
Now, substituting \( x^2 = a^2 \cos(2\theta) \) into the integral:
\[ \sqrt{a^2 - x^2} = \sqrt{a^2 - a^2 \cos(2\theta)} = \sqrt{a^2(1 - \cos(2\theta))} = a \sqrt{2 \sin^2(\theta)} = a \sqrt{2} \sin(\theta) \]
And,
\[ \sqrt{a^2 + x^2} = \sqrt{a^2 + a^2 \cos(2\theta)} = \sqrt{a^2(1 + \cos(2\theta))} = a \sqrt{2 \cos^2(\theta)} = a \sqrt{2} \cos(\theta) \]

### Step 3: Substitute into the Integral
Substituting everything into the integral:
\[ \int x \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} \, dx = \int -\frac{a^2}{2} \sin(2\theta) \cdot \frac{a \sqrt{2} \sin(\theta)}{a \sqrt{2} \cos(\theta)} \, d\theta \]
This simplifies to:
\[ -\frac{a^3}{2} \int \sin(2\theta) \tan(\theta) \, d\theta \]

### Step 4: Simplifying the Integral
Using the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we can rewrite the integral:
\[ -\frac{a^3}{2} \int 2 \sin(\theta) \cos(\theta) \tan(\theta) \, d\theta = -a^3 \int \sin^2(\theta) \, d\theta \]

### Step 5: Integrate
The integral of \( \sin^2(\theta) \) can be evaluated using the identity \( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \):
\[ -a^3 \int \sin^2(\theta) \, d\theta = -a^3 \left( \frac{1}{2} \theta - \frac{1}{4} \sin(2\theta) \right) + C \]

### Step 6: Back Substitution
Now we need to substitute back for \( \theta \):
From \( x^2 = a^2 \cos(2\theta) \), we have \( \cos(2\theta) = \frac{x^2}{a^2} \) and hence \( \theta = \frac{1}{2} \cos^{-1}\left(\frac{x^2}{a^2}\right) \).

Also, \( \sin(2\theta) = 1 - \cos^2(2\theta) = 1 - \frac{x^4}{a^4} \).

### Final Result
Putting everything together, we get:
\[ \int x \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} \, dx = -\frac{a^3}{4} \cos^{-1}\left(\frac{x^2}{a^2}\right) + \frac{a^3}{8} \left(1 - \frac{x^4}{a^4}\right) + C \]