Question

Evaluate: \[ \int_{0}^{\pi} \sqrt{1+4\cos\frac{x}{2}} \left(\cos\frac{x}{2}-1\right)\,dx \]

• A. \[ 4\sqrt3-4-\frac{\pi}{3} \]
• B. \[ 4\sqrt3-4-\frac{4\pi}{3} \]
• C. \[ \frac{4\pi}{3}-4\sqrt3+4 \]
• D. \[ \frac{\pi}{3}-4\sqrt3+4 \]

Hint:

Whenever a definite integral contains \[ \sqrt{a+b\cos\frac{x}{2}}, \] try expressing the square root itself as a new variable. This often transforms a complicated trigonometric integral into an algebraic integral with manageable limits.

Answer 1

The Correct Option is A