Question

Evaluate: \[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \,dx. \]

Hint:

Use the identity \( \cos^2 x = \frac{1 + \cos 2x}{2} \) for integrals involving even powers of trigonometric functions.

Answer 1

Step 1: Use the Power Reduction Formula
\[ \cos^2 x = \frac{1 + \cos 2x}{2}. \] Step 2: Substitute into the Integral
\[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \,dx = \int_{0}^{\frac{\pi}{2}} \frac{1 + \cos 2x}{2} dx. \] Splitting: \[ = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} dx + \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \cos 2x \,dx. \] Step 3: Evaluate Each Integral
\[ \frac{1}{2} \left[ x \right]_{0}^{\frac{\pi}{2}} + \frac{1}{2} \left[ \frac{\sin 2x}{2} \right]_{0}^{\frac{\pi}{2}}. \] \[ = \frac{1}{2} \times \frac{\pi}{2} + \frac{1}{2} \times 0. \] \[ = \frac{\pi}{4}. \]