If $\int\limits_{0}^{2\pi} \cos^2 x \, dx = k \int\limits_{0}^{\pi/2} \cos^2 x \, dx$, then the value of $k$ is:

4

The correct answer is Option (1) → 4

$\int\limits_{0}^{2\pi} \cos^2 x \, dx = k \int\limits_{0}^{\pi/2} \cos^2 x \, dx \text{}$

Let $f(x) = \cos^2 x$

$f(2\pi - x) = \cos^2(2\pi - x) = \cos^2 x \text{}$

$∴\int\limits_{0}^{2\pi} \cos^2 x \, dx = 2 \int\limits_{0}^{\pi} \cos^2 x \, dx \text{}$

Similarly, $\int\limits_{0}^{\pi} \cos^2 x \, dx = 2 \times \int\limits_{0}^{\pi/2} \cos^2 x \, dx \text{}$

$∴\int\limits_{0}^{2\pi} \cos^2 x \, dx = 4 \int\limits_{0}^{\pi/2} \cos^2 x \, dx \text{}$

$∴k = 4 \text{}$