Step 1: Understanding the Concept:
This problem involves solving a complex-looking integral by identifying a derivative of a quotient and then analyzing the resulting trigonometric equation with a fractional part function.
Step 2: Key Formula or Approach:
Consider the derivative of \( \frac{\cot x}{(\cos x)^n} \).
\[ \frac{d}{dx} \left( \frac{\cot x}{(\cos x)^{2010}} \right) = \frac{(-\csc^2 x)(\cos x)^{2010} - (\cot x)(2010)(\cos x)^{2009}(-\sin x)}{(\cos x)^{4020}} \]
Simplify the second term in the numerator: \( \cot x \cdot \sin x = \cos x \).
\[ = \frac{-\csc^2 x (\cos x)^{2010} + 2010 (\cos x)^{2010}}{(\cos x)^{4020}} = \frac{-\csc^2 x + 2010}{(\cos x)^{2010}} \]
Step 3: Detailed Explanation:
The integral is:
\[ \int \frac{\csc^2 x - 2010}{\cos^{2010} x} dx = - \int \frac{-\csc^2 x + 2010}{\cos^{2010} x} dx \]
From our derivative calculation:
\[ = - \left( \frac{\cot x}{(\cos x)^{2010}} \right) + c \]
Comparing this with the given form \( -\frac{f(x)}{(g(x))^{2010}} \), we identify:
- \( f(x) = \cot x \)
- \( g(x) = \cos x \)
Check condition: \( f(\pi/4) = \cot(\pi/4) = 1 \). Correct.
Now we solve the equation:
\[ \frac{f(x)}{g(x)} = \{x\} \implies \frac{\cot x}{\cos x} = \{x\} \implies \frac{\cos x}{\sin x \cos x} = \{x\} \implies \csc x = \{x\} \]
Analyze the range:
- For any real \( x \), \( \{x\} \in [0, 1) \).
- The function \( \csc x \) has a range \( (-\infty, -1] \cup [1, \infty) \).
There is no overlap between the set \( [0, 1) \) and the set \( (-\infty, -1] \cup [1, \infty) \).
Step 4: Final Answer:
Since the range of the Left-Hand Side and Right-Hand Side are disjoint, there are no solutions. Thus, the number of solutions is 0.