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:
Answer & explanation
Correct answer: option 1
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, $\int\limits_{0}^{\pi} \cos^2 x \, dx = 2 \times \int\limits_{0}^{\pi/2} \cos^2 x \, dx \text{}$
$∴\int\limits_{0}^{2\pi} \cos^2 x \, dx = 4 \int\limits_{0}^{\pi/2} \cos^2 x \, dx \text{}$
$∴k = 4 \text{}$