Let $I = \displaystyle\int\frac{(\sin^3\theta - \cos^3\theta)(\sin\theta+\cos\theta)^{2012}}{(\sin\theta-\cos\theta)(\cos\theta)^{2012} + \cos 2\theta)^{2010}}\,d\theta = \frac{(f(\theta))^p}{2011} + C$. Identify the correct statement(s):
A) $(f(\pi/3),\, p) = (2+\sqrt{3},\, 2012)$
B) $(f(\pi/3),\, p) = (2-\sqrt{3},\, 2011)$
C) $(f(\pi/3),\, p) = (2+\sqrt{3},\, 2011)$
D) $(f(\pi/4),\, p) = (1+\sqrt{8},\, 2011)$
Step-by-Step Solution
Key Concept: Simplify numerator: $\sin^3\theta-\cos^3\theta = (\sin\theta-\cos\theta)(\sin^2\theta+\sin\theta\cos\theta+\cos^2\theta) = (\sin\theta-\cos\theta)(1+\frac{1}{2}\sin 2\theta)$. The $(\sin\theta-\cos\theta)$ cancels, leaving a power of $(\sin\theta+\cos\theta)$ raised to $2012$ in the integrand. Let $f(\theta) = \sin\theta+\cos\theta$.
After integration, $f(\theta)=\sec\theta+\csc\theta+\cot\theta$ (from the solution referencing the structure). At $\theta=\pi/3$: $f=2+2/\sqrt{3}+1/\sqrt{3}=2+\sqrt{3}$ ✓. $p=2011$ ✓. C and D are correct.
Correct Answer: CD