The value of `int_(0)^(pi//2) (sin 8x log cot x)/(cos 2x)dx`, is

Options

A. 0
B. `pi`
C. `(5pi)/(2)`
D. `(3pi)/(2)`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin(8x) \log(\cot x)}{\cos(2x)} \, dx \), we can use the property of definite integrals. The property states that:

\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a - x) \, dx \]

### Step 1: Apply the property of definite integrals

Let's apply this property to our integral:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin(8x) \log(\cot x)}{\cos(2x)} \, dx \]

By substituting \( x \) with \( \frac{\pi}{2} - x \):

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin\left(8\left(\frac{\pi}{2} - x\right)\right) \log\left(\cot\left(\frac{\pi}{2} - x\right)\right)}{\cos\left(2\left(\frac{\pi}{2} - x\right)\right)} \, dx \]

### Step 2: Simplify the terms

Now, simplify each term:

1. \( \sin\left(8\left(\frac{\pi}{2} - x\right)\right) = \sin\left(4\pi - 8x\right) = -\sin(8x) \)
2. \( \log\left(\cot\left(\frac{\pi}{2} - x\right)\right) = \log(\tan x) \)
3. \( \cos\left(2\left(\frac{\pi}{2} - x\right)\right) = \cos(\pi - 2x) = -\cos(2x) \)

Substituting these into the integral gives:

\[ I = \int_{0}^{\frac{\pi}{2}} \frac{-\sin(8x) \log(\tan x)}{-\cos(2x)} \, dx = \int_{0}^{\frac{\pi}{2}} \frac{\sin(8x) \log(\tan x)}{\cos(2x)} \, dx \]

### Step 3: Combine the integrals

Now we have two expressions for \( I \):

1. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin(8x) \log(\cot x)}{\cos(2x)} \, dx \)
2. \( I = \int_{0}^{\frac{\pi}{2}} \frac{\sin(8x) \log(\tan x)}{\cos(2x)} \, dx \)

Adding these two equations:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin(8x) \left(\log(\cot x) + \log(\tan x)\right)}{\cos(2x)} \, dx \]

Using the logarithmic identity \( \log(a) + \log(b) = \log(ab) \):

\[ \log(\cot x) + \log(\tan x) = \log(1) = 0 \]

Thus, we have:

\[ 2I = \int_{0}^{\frac{\pi}{2}} \frac{\sin(8x) \cdot 0}{\cos(2x)} \, dx = 0 \]

### Step 4: Solve for \( I \)

This implies:

\[ I = 0 \]

### Conclusion

The value of the integral \( \int_{0}^{\frac{\pi}{2}} \frac{\sin(8x) \log(\cot x)}{\cos(2x)} \, dx \) is \( 0 \).