Definite Integration
Integral Calculus-2
star_batch_jee_advanced_2025
Grade 12
Question:
If $\int_0^\pi \sqrt{(\cos x + \cos 2x + \cos 3x)^2 + (\sin x + \sin 2x + \sin 3x)^2} dx$ has the value equal to $\left(\frac{\pi}{k} + \sqrt{w}\right)$ where $k$ and $w$ are positive integers then $k^2 + w^2 = $ ____.
Step-by-Step Solution
Key Concept: Recognize the nested trigonometric expression simplifies to a perfect square after using sum-to-product identities.
Simplify the integrand: $\sqrt{3+2(\cos x + \cos x + \cos 2x)} = \sqrt{4\cos^2 x + 4\cos x + 1} = |1 + 2\cos x|$. Split the integral at $x = \frac{2\pi}{3}$ where $1 + 2\cos x = 0$. Integrate $\int_0^{2\pi/3} (1+2\cos x)dx$ and $\int_{2\pi/3}^\pi (1+2\cos x)dx$ separately to obtain $\frac{\pi}{3} + 2\sqrt{3} + \frac{\pi}{3} + \sqrt{12} = \frac{2\pi}{3} + 2\sqrt{3} + \sqrt{12}$. Thus $k=3, w=12$ and $k^2 + w^2 = 153$.
Correct Answer: 153