Question

Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx \)

• A. 2
• B. 0
• C. -1
• D. 5

Hint:

For integrals of even functions over symmetric limits, double the integral over half of the interval.

Answer 1

The Correct Option is A

The integral we need to solve is: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx \] Since \( \cos x \) is an even function (i.e., \( \cos(-x) = \cos(x) \)), the integral of \( \cos x \) over symmetric limits cancels out: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x \, dx = 2 \int_0^{\frac{\pi}{2}} \cos x \, dx \] The integral of \( \cos x \) is \( \sin x \), so: \[ 2 [ \sin x ]_0^{\frac{\pi}{2}} = 2 (1 - 0) = 2 \] Thus, the correct answer is 2.