Let `f(x)=7 tan^(8) x +7 tan^(6) x-3 tan^(4) x-3 tan^(2)x ` for all `x in(-(pi)/(2),(pi)/(2))`. Then the correct expression (s) is are

Options

A. ` int_(0)^(pi//4)xf(x) dx =(1)/(12)`
B. ` int_(0)^(pi//4)f(x) dx =0`
C. ` int_(0)^(pi//4)xf(x) dx =(1)/(6)`
D. ` int_(0)^(pi//4)f(x) dx =1`

Answer

To solve the problem, we need to evaluate the function \( f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x \) and find the correct expressions based on the given options.

### Step 1: Rewrite the function
We can rewrite the function \( f(x) \) as follows:
\[ f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x \]

### Step 2: Factor out common terms
Notice that we can factor out \( \tan^2 x \) from the expression:
\[ f(x) = \tan^2 x \left( 7 \tan^6 x + 7 \tan^4 x - 3 \tan^2 x - 3 \right) \]

### Step 3: Substitute \( t = \tan^2 x \)
Let \( t = \tan^2 x \). Then the function becomes:
\[ f(x) = t \left( 7t^3 + 7t^2 - 3t - 3 \right) \]

### Step 4: Analyze the polynomial
Now we need to analyze the polynomial \( 7t^3 + 7t^2 - 3t - 3 \). We can check for roots or use the Rational Root Theorem to find possible rational roots.

### Step 5: Finding the roots
By testing values, we find that \( t = 1 \) is a root:
\[ 7(1)^3 + 7(1)^2 - 3(1) - 3 = 7 + 7 - 3 - 3 = 8 \quad \text{(not a root)} \]
Testing \( t = -1 \):
\[ 7(-1)^3 + 7(-1)^2 - 3(-1) - 3 = -7 + 7 + 3 - 3 = 0 \quad \text{(is a root)} \]

### Step 6: Polynomial division
Now we can perform polynomial long division to factor \( 7t^3 + 7t^2 - 3t - 3 \) by \( t + 1 \).

### Step 7: Factor the polynomial
After performing the division, we find:
\[ 7t^3 + 7t^2 - 3t - 3 = (t + 1)(7t^2 + 0t - 3) \]

### Step 8: Solve the quadratic
Now we can solve \( 7t^2 - 3 = 0 \):
\[ t^2 = \frac{3}{7} \implies t = \sqrt{\frac{3}{7}} \quad \text{(since \( t = \tan^2 x \) must be non-negative)} \]

### Step 9: Substitute back to find \( x \)
Thus, we have:
\[ \tan^2 x = 1 \quad \text{or} \quad \tan^2 x = \frac{3}{7} \]
This gives us:
\[ x = \frac{\pi}{4} \quad \text{or} \quad x = \tan^{-1}\left(\sqrt{\frac{3}{7}}\right) \]

### Conclusion
The correct expressions for \( f(x) \) based on the evaluation and roots found are:
- \( x = \frac{\pi}{4} \)
- \( x = \tan^{-1}\left(\sqrt{\frac{3}{7}}\right) \)