If `int_(-pi//3)^(pi//3) [(a)/(3) |tan x| + (b tan x)/(1+sec x)+ c] dx = 0`, where a, b, c are constants, then c=

Options

A. `-(a)/(pi) log 2`
B. `(2a)/(pi) log 2`
C. `a log 2`
D. `(a)/(pi) log 2`

Answer

To solve the given integral equation

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \left( \frac{a}{3} |\tan x| + \frac{b \tan x}{1 + \sec x} + c \right) dx = 0, \]

where \( a, b, c \) are constants, we will break the integral into three parts and analyze each part.

### Step 1: Break the integral into three parts

We can express the integral as:

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{a}{3} |\tan x| \, dx + \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{b \tan x}{1 + \sec x} \, dx + \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} c \, dx = 0. \]

### Step 2: Evaluate the first integral

The first integral involves \( |\tan x| \). Since \( \tan x \) is an odd function, \( |\tan x| \) is an even function. Therefore, we can write:

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} |\tan x| \, dx = 2 \int_{0}^{\frac{\pi}{3}} \tan x \, dx. \]

Calculating the integral:

\[ \int \tan x \, dx = -\log |\cos x| + C. \]

Thus,

\[ \int_{0}^{\frac{\pi}{3}} \tan x \, dx = -\log |\cos(\frac{\pi}{3})| + \log |\cos(0)| = -\log \left(\frac{1}{2}\right) + \log(1) = \log(2). \]

So,

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} |\tan x| \, dx = 2 \log(2). \]

Therefore,

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{a}{3} |\tan x| \, dx = \frac{2a}{3} \log(2). \]

### Step 3: Evaluate the second integral

For the second integral, we have:

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{b \tan x}{1 + \sec x} \, dx. \]

Since \( \tan(-x) = -\tan(x) \) and \( \sec(-x) = \sec(x) \), the function \( \frac{\tan x}{1 + \sec x} \) is odd. Therefore,

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{b \tan x}{1 + \sec x} \, dx = 0. \]

### Step 4: Evaluate the third integral

The third integral is straightforward:

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} c \, dx = c \left( \frac{\pi}{3} - \left(-\frac{\pi}{3}\right) \right) = c \cdot \frac{2\pi}{3}. \]

### Step 5: Combine all parts

Now we combine all parts:

\[ \frac{2a}{3} \log(2) + 0 + c \cdot \frac{2\pi}{3} = 0. \]

### Step 6: Solve for \( c \)

Rearranging gives:

\[ c \cdot \frac{2\pi}{3} = -\frac{2a}{3} \log(2). \]

Dividing both sides by \( \frac{2\pi}{3} \):

\[ c = -\frac{a}{\pi} \log(2). \]

### Final Answer

Thus, the value of \( c \) is:

\[ \boxed{-\frac{a}{\pi} \log(2)}. \]