If `int tan^(7) x dx = f(x) + C` then

Options

A. f(x) is a polynomial of degree 8 in tan x
B. f(x) is a polynomial of degree 5 in tan x
C. f(x) `= (1)/(6) tan^(6) x - (1)/(4) tan^(4)x + (1)/(2)sec^(2) x + log | cos x| + C`
D. f(x) is a polynomial of degree 6 in tan x

Answer

To solve the integral \( \int \tan^7 x \, dx \), we will use the reduction formula for integrals of the form \( \int \tan^n x \, dx \).

### Step-by-step Solution:

1. **Set up the integral**:
Let \( I = \int \tan^7 x \, dx \).

2. **Apply the reduction formula**:
The reduction formula for \( \tan^n x \) when \( n \geq 2 \) is given by:
\[ \int \tan^n x \, dx = \frac{1}{n-1} \tan^{n-1} x - \int \tan^{n-2} x \, dx \]
For \( n = 7 \):
\[ I = \frac{1}{6} \tan^6 x - \int \tan^5 x \, dx \]

3. **Evaluate \( \int \tan^5 x \, dx \)**:
Let \( J = \int \tan^5 x \, dx \).
Using the reduction formula again for \( n = 5 \):
\[ J = \frac{1}{4} \tan^4 x - \int \tan^3 x \, dx \]

4. **Evaluate \( \int \tan^3 x \, dx \)**:
Let \( K = \int \tan^3 x \, dx \).
Again applying the reduction formula for \( n = 3 \):
\[ K = \frac{1}{2} \tan^2 x - \int \tan x \, dx \]

5. **Evaluate \( \int \tan x \, dx \)**:
The integral of \( \tan x \) is:
\[ \int \tan x \, dx = -\log |\cos x| + C \]

6. **Combine results**:
Now substitute back:
\[ K = \frac{1}{2} \tan^2 x + \log |\cos x| + C \]
Substitute \( K \) into \( J \):
\[ J = \frac{1}{4} \tan^4 x - \left( \frac{1}{2} \tan^2 x + \log |\cos x| \right) \]
Simplifying gives:
\[ J = \frac{1}{4} \tan^4 x - \frac{1}{2} \tan^2 x - \log |\cos x| + C \]

7. **Substitute \( J \) into \( I \)**:
Substitute \( J \) into \( I \):
\[ I = \frac{1}{6} \tan^6 x - \left( \frac{1}{4} \tan^4 x - \frac{1}{2} \tan^2 x - \log |\cos x| \right) \]
This simplifies to:
\[ I = \frac{1}{6} \tan^6 x - \frac{1}{4} \tan^4 x + \frac{1}{2} \tan^2 x + \log |\cos x| + C \]

8. **Final expression**:
Therefore, the result of the integral is:
\[ \int \tan^7 x \, dx = \frac{1}{6} \tan^6 x - \frac{1}{4} \tan^4 x + \frac{1}{2} \tan^2 x + \log |\cos x| + C \]

### Conclusion:
The correct option from the given choices is:
\[ f(x) = \frac{1}{6} \tan^6 x - \frac{1}{4} \tan^4 x + \frac{1}{2} \sec^2 x + \log |\cos x| + C \]