The value of the integral `int_(pi//3)^(pi//3) (x sinx)/(cos^(2)x)dx`, is

Options

A. `(pi//3-logtan3pi//2)`
B. `2(2pi//3-logtan5pi//12)`
C. `3(pi//2-logsinpi//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 will use integration by parts.

### Step 1: Rewrite the integral
We can rewrite \(\cos^2 x\) as \(\cos x \cdot \cos x\), so we have:

\[ I = \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} x \sin x \sec^2 x \, dx. \]

### Step 2: Apply integration by parts
Let \(u = x\) and \(dv = \sin x \sec^2 x \, dx\). Then, we differentiate and integrate:

\[ du = dx, \quad v = \int \sin x \sec^2 x \, dx. \]

We know that \(\sec^2 x = 1 + \tan^2 x\) and \(\sin x = \tan x \cos x\), so:

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

### Step 3: Set up integration by parts formula
Using the integration by parts formula \(\int u \, dv = uv - \int v \, du\):

\[ I = \left[ x \left(-\log |\cos x|\right) \right]_{-\frac{\pi}{3}}^{\frac{\pi}{3}} - \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} -\log |\cos x| \, dx. \]

### Step 4: Evaluate the boundary terms
Now we evaluate the boundary terms:

\[ \left[ -x \log |\cos x| \right]_{-\frac{\pi}{3}}^{\frac{\pi}{3}} = -\left(\frac{\pi}{3} \log |\cos(\frac{\pi}{3})| - \left(-\frac{\pi}{3} \log |\cos(-\frac{\pi}{3})|\right)\right). \]

Since \(\cos(\frac{\pi}{3}) = \cos(-\frac{\pi}{3}) = \frac{1}{2}\):

\[ = -\left(\frac{\pi}{3} \log \frac{1}{2} + \frac{\pi}{3} \log \frac{1}{2}\right) = -\left(\frac{2\pi}{3} \log \frac{1}{2}\right) = \frac{2\pi}{3} \log 2. \]

### Step 5: Evaluate the integral of \(-\log |\cos x|\)
Next, we need to evaluate:

\[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \log |\cos x| \, dx. \]

This integral is known and can be evaluated as:

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

### Step 6: Combine results
Now substituting back into our expression for \(I\):

\[ I = \frac{2\pi}{3} \log 2 + \frac{\pi}{3} \log 2 = \frac{3\pi}{3} \log 2 = \pi \log 2. \]

Thus, the value of the integral is:

\[ \boxed{\pi \log 2}. \]