Indefinite Integration
Trigonometric Integration
Grade 12
Question:
<p>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 \(\{\}\) represents fractional part function)</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Integration combined with fractional part function analysis requires careful evaluation of the integral and graphical/algebraic solution of the resulting equation.
<p>Evaluate the integral to find $f(x)$ and $g(x)$, then use the boundary condition $f(\pi/4) = 1$ to determine the functions. Finally, solve $f(x)/g(x) = \{x\}$ graphically or algebraically in the given interval.</p>
Correct Answer: D