Question

Evaluate the integral: \[ \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx = ? \]

• A. \( 2 \sin \sqrt{x} + c \)
• B. \( \sin \sqrt{x} + c \)
• C. \( \cos \sqrt{x} + c \)
• D. \( 2 \cos \sqrt{x} + c \)

Hint:

Use substitution \( t = \sqrt{x} \) to simplify integrals involving \( \cos \sqrt{x} \) divided by \( \sqrt{x} \).

Answer 1

The Correct Option is A

Let: \[ t = \sqrt{x} \implies x = t^2, \quad dx = 2t \, dt \] Substitute in the integral: \[ \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx = \int \frac{\cos t}{t} \cdot 2t \, dt = \int 2 \cos t \, dt = 2 \sin t + c = 2 \sin \sqrt{x} + c \]