`I= int_(-pi//3)^(pi//3) (x sin x)/(cos^(2)x) dx` is equal to

Options

A. `((pi)/(3)- log "tan"(3pi)/(2))`
B. `2((2pi)/(3)- log "tan"(5pi)/(12))`
C. `3((pi)/(2)- "log sin"(5pi)/(12))`
D. None of these

Answer

To solve the integral \( I = \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{x \sin x}{\cos^2 x} \, dx \), we can use the properties of definite integrals.

### Step 1: Identify the function
Let \( f(x) = \frac{x \sin x}{\cos^2 x} \).

### Step 2: Check the symmetry of the function
We need to check if \( f(-x) = -f(x) \) or \( f(-x) = f(x) \).

Calculating \( f(-x) \):
\[ f(-x) = \frac{-x \sin(-x)}{\cos^2(-x)} = \frac{-x (-\sin x)}{\cos^2 x} = \frac{x \sin x}{\cos^2 x} = f(x) \]
Since \( f(-x) = f(x) \), the function is even.

### Step 3: Apply the property of definite integrals
Since \( f(x) \) is even, we can use the property:
\[ \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx \]
Thus,
\[ I = 2 \int_{0}^{\frac{\pi}{3}} \frac{x \sin x}{\cos^2 x} \, dx \]

### Step 4: Rewrite the integrand
We can rewrite the integrand:
\[ \frac{x \sin x}{\cos^2 x} = x \tan x \sec x \]
So,
\[ I = 2 \int_{0}^{\frac{\pi}{3}} x \tan x \sec x \, dx \]

### Step 5: Integration by parts
Let \( u = x \) and \( dv = \tan x \sec x \, dx \).
Then, \( du = dx \) and \( v = \sec x \).

Using integration by parts:
\[ \int u \, dv = uv - \int v \, du \]
We have:
\[ I = 2 \left[ x \sec x \bigg|_{0}^{\frac{\pi}{3}} - \int_{0}^{\frac{\pi}{3}} \sec x \, dx \right] \]

### Step 6: Evaluate the boundary term
Calculating the boundary term:
\[ x \sec x \bigg|_{0}^{\frac{\pi}{3}} = \left( \frac{\pi}{3} \sec\left(\frac{\pi}{3}\right) - 0 \cdot \sec(0) \right) \]
Since \( \sec\left(\frac{\pi}{3}\right) = 2 \):
\[ = \frac{\pi}{3} \cdot 2 = \frac{2\pi}{3} \]

### Step 7: Evaluate the integral of sec x
The integral of \( \sec x \) is:
\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C \]
Thus,
\[ \int_{0}^{\frac{\pi}{3}} \sec x \, dx = \ln \left| \sec\left(\frac{\pi}{3}\right) + \tan\left(\frac{\pi}{3}\right) \right| - \ln \left| \sec(0) + \tan(0) \right| \]
Calculating:
\[ \sec\left(\frac{\pi}{3}\right) = 2, \quad \tan\left(\frac{\pi}{3}\right) = \sqrt{3}, \quad \sec(0) = 1, \quad \tan(0) = 0 \]
Thus,
\[ \int_{0}^{\frac{\pi}{3}} \sec x \, dx = \ln(2 + \sqrt{3}) - \ln(1) = \ln(2 + \sqrt{3}) \]

### Step 8: Combine results
Putting it all together:
\[ I = 2 \left( \frac{2\pi}{3} - \ln(2 + \sqrt{3}) \right) \]
Thus,
\[ I = \frac{4\pi}{3} - 2\ln(2 + \sqrt{3}) \]

### Final Answer
\[ I = \frac{4\pi}{3} - 2\ln(2 + \sqrt{3}) \]