If `int tan^(5)xex=Atan^(4)x+Btan^(x)+g(x)+C`, where C is constant of integration and `g(0)=0, ` then

Options

A. `A=1/4, B=- 1/2`
B. `g(x)= ln |sec x|`
C. `g(x)= ln |cos x|`
D. `A= - 1/4, B=1/3`

Answer

To solve the integral \( \int \tan^5 x \, dx \) and express it in the form \( A \tan^4 x + B \tan x + g(x) + C \), where \( C \) is the constant of integration and \( g(0) = 0 \), we can follow these steps:

### Step-by-Step Solution

1. **Rewrite the Integral**:
\[ \int \tan^5 x \, dx = \int \tan^3 x \tan^2 x \, dx = \int \tan^3 x (\sec^2 x - 1) \, dx \]
Here, we used the identity \( \tan^2 x = \sec^2 x - 1 \).

**Hint**: Use trigonometric identities to simplify the integral.

2. **Split the Integral**:
\[ = \int \tan^3 x \sec^2 x \, dx - \int \tan^3 x \, dx \]

**Hint**: Splitting the integral can help in managing the terms separately.

3. **Substitution for the First Integral**:
Let \( t = \tan x \), then \( dt = \sec^2 x \, dx \). Thus, the first integral becomes:
\[ \int t^3 \, dt = \frac{t^4}{4} = \frac{\tan^4 x}{4} \]

**Hint**: Use substitution to convert the integral into a polynomial form.

4. **Substituting for the Second Integral**:
For the second integral \( \int \tan^3 x \, dx \), we can rewrite it as:
\[ \int \tan x \tan^2 x \, dx = \int \tan x (\sec^2 x - 1) \, dx = \int \tan x \sec^2 x \, dx - \int \tan x \, dx \]

**Hint**: Recognize that you can express \( \tan^2 x \) in terms of \( \sec^2 x \).

5. **Evaluate the First Part**:
For \( \int \tan x \sec^2 x \, dx \), using substitution \( t = \tan x \):
\[ \int t \, dt = \frac{t^2}{2} = \frac{\tan^2 x}{2} \]

**Hint**: Keep applying substitution to simplify the integral.

6. **Evaluate the Second Part**:
The integral \( \int \tan x \, dx \) is:
\[ -\log |\cos x| + C \]

**Hint**: Remember the integral of \( \tan x \) is a standard result.

7. **Combine All Parts**:
Now, putting everything together:
\[ \int \tan^5 x \, dx = \frac{\tan^4 x}{4} - \left( \frac{\tan^2 x}{2} + (-\log |\cos x|) \right) + C \]
Simplifying gives:
\[ = \frac{\tan^4 x}{4} - \frac{\tan^2 x}{2} + \log |\sec x| + C \]

**Hint**: Combine like terms and simplify to get the final expression.

8. **Identify Constants**:
From the expression, we can identify:
\[ A = \frac{1}{4}, \quad B = -\frac{1}{2}, \quad g(x) = \log |\sec x| \]

**Hint**: Compare coefficients to find the values of \( A \) and \( B \).

9. **Check Condition \( g(0) = 0 \)**:
Since \( g(0) = \log |\sec(0)| = \log |1| = 0 \), the condition is satisfied.

**Hint**: Always verify conditions given in the problem.

### Final Answer
Thus, we have:
- \( A = \frac{1}{4} \)
- \( B = -\frac{1}{2} \)
- \( g(x) = \log |\sec x| \)