Question

(c) The value of the integral \( \int \sqrt{1 + \sin 2x} \,dx \) is:

• A. \( \sin x + \cos x + c \)
• B. \( \sin x - \cos x + c \)
• C. \( \cos x - \sin x + c \)
• D. \( \cos x + \sin x + c \)

Hint:

Use trigonometric identities for simplifying integrals before integrating.

Answer 1

The Correct Option is B

Step 1: Use trigonometric identity \( \sin 2x = 2 \sin x \cos x \). Step 2: Rewrite the integral using substitution: \[ \int \sqrt{1 + \sin 2x} \,dx = \int (\sin x - \cos x) \,dx \] Step 3: Integrate each term. \[ \int \sin x \,dx = -\cos x, \quad \int \cos x \,dx = \sin x \] Thus, \[ \sin x - \cos x + c. \]