Question

If \(\int\frac{\csc^{2}x-2010}{\cos^{2010}x}dx=-\frac{f(x)}{(g(x))^{2010}}+c\), where \(f\left(\frac{\pi}{4}\right)=1\), then the number of solutions of the equation \(\frac{f(x)}{g(x)}=\{x\}\) in \([0,2\pi]\) is/are (where \(\{\cdot\}\) represents fractional part function):

• A. 3
• B. 1
• C. 0
• D. 2

Hint:

Whenever an integration problem contains a large year number (like 2010), don't panic! It is a sign that the problem is designed around a clean cancellation pattern or power derivative rule where the large number acts as a constant multiplier.

Answer 1

The Correct Option is C

Concept: This problem requires integrating a complex trigonometric expression by recognizing it as a reverse product rule expansion. Once the functions $f(x)$ and $g(x)$ are isolated, we can evaluate the number of intersecting solutions by comparing the numerical output ranges of both sides of the equation. Step 1: Deconstruct the integrand using exponents.
Let us rewrite the given integral expression by converting the fractions into negative power exponents: $$I = \int \frac{\csc^2 x - 2010}{\cos^{2010}x} \, dx = \int \left( \csc^2 x \cdot \cos^{-2010}x - 2010\cos^{-2010}x \right) dx$$ Let us test the derivative of a candidate product function structured like the answer template, $\frac{\cot x}{\cos^{2010}x} = \cot x \cdot \cos^{-2010}x$: $$\frac{d}{dx}\left[ \cot x \cdot \cos^{-2010}x \right] = \left(-\csc^2 x\right)\cos^{-2010}x + \cot x \cdot \left(-2010\cos^{-2011}x \cdot (-\sin x)\right)$$ Simplify the second term by breaking down the cotangent function ($\cot x = \frac{\cos x}{\sin x}$): $$= -\csc^2 x \cdot \cos^{-2010}x + \left(\frac{\cos x}{\sin x}\right) \cdot 2010 \cdot \cos^{-2011}x \cdot \sin x$$ Notice that the sine terms cancel out perfectly, and combining the cosine bases ($\cos x \cdot \cos^{-2011}x = \cos^{-2010}x$) gives: $$\frac{d}{dx}\left[ \frac{\cot x}{\cos^{2010}x} \right] = -\frac{\csc^2 x}{\cos^{2010}x} + \frac{2010}{\cos^{2010}x} = -\left( \frac{\csc^2 x - 2010}{\cos^{2010}x} \right)$$

Step 2: Isolate the functions $f(x)$ and $g(x)$.
By integrating both sides of our derived differential identity, we find: $$\int \frac{\csc^2 x - 2010}{\cos^{2010}x} \, dx = -\frac{\cot x}{\cos^{2010}x} + c$$ Comparing this directly with the given answer template $-\frac{f(x)}{(g(x))^{2010}} + c$, we can extract our primary functions: $$f(x) = \cot x \quad \text{and} \quad g(x) = \cos x$$ Let us double-check the initial condition: $f\left(\frac{\pi}{4}\right) = \cot\left(\frac{\pi}{4}\right) = 1$, which matches the problem statement perfectly.

Step 3: Set up the target cross-over equation.
Form the ratio of the two functions as requested by the equation: $$\frac{f(x)}{g(x)} = \frac{\cot x}{\cos x} = \frac{\left(\frac{\cos x}{\sin x}\right)}{\cos x} = \frac{1}{\sin x} = \csc x$$ The problem reduces to finding the number of solutions for: $$\csc x = \{x\} \quad \text{for } x \in [0, 2\pi]$$

Step 4: Analyze the intersection of their output ranges.
Let us compare the mathematical boundaries for both functions:
• The fractional part function $\{x\}$ filters out integers and is strictly bounded by definition: $0 \le \{x\} < 1$.
• The cosecant function $\csc x$ can only output values in the range: $(-\infty, -1] \cup [1, \infty)$. Because the minimum positive value of $\csc x$ is $1$ and the maximum possible value of $\{x\}$ is strictly less than $1$, the two graphs can never cross paths. This means there are 0 solutions inside the domain.