Evaluate : (i) `int_(-pi//2)^(pi//2)(g(x)-g(-x))/(f(-x)+f(x))`

Answer

To evaluate the integral

\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{g(x) - g(-x)}{f(-x) + f(x)} \, dx, \]

we can use properties of definite integrals.

### Step 1: Change of Variables
Let's first perform a change of variables in the integral. We can substitute \( x \) with \( -x \):

\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{g(-x) - g(x)}{f(x) + f(-x)} \, (-dx). \]

This changes the limits of integration, but since the limits are symmetric about zero, we can rewrite it as:

\[ I = -\int_{\frac{\pi}{2}}^{-\frac{\pi}{2}} \frac{g(-x) - g(x)}{f(x) + f(-x)} \, dx = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{g(-x) - g(x)}{f(x) + f(-x)} \, dx. \]

### Step 2: Combine the Integrals
Now we have two expressions for \( I \):

1. \( I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{g(x) - g(-x)}{f(-x) + f(x)} \, dx \)
2. \( I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{g(-x) - g(x)}{f(x) + f(-x)} \, dx \)

Adding these two equations gives us:

\[ 2I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{g(x) - g(-x)}{f(-x) + f(x)} + \frac{g(-x) - g(x)}{f(x) + f(-x)} \right) \, dx. \]

### Step 3: Simplify the Expression
Notice that the two fractions in the integral have the same denominator \( f(-x) + f(x) \). The numerators combine as follows:

\[ g(x) - g(-x) + g(-x) - g(x) = 0. \]

Thus, we have:

\[ 2I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 0 \, dx = 0. \]

### Step 4: Solve for I
From the equation \( 2I = 0 \), we can solve for \( I \):

\[ I = 0. \]

### Final Answer
Therefore, the value of the integral is

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{g(x) - g(-x)}{f(-x) + f(x)} \, dx = 0. \]

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