Question

Evaluate \( \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx \)

• A. \( \frac{1}{2} \cos \sqrt{x} + c \)
• B. \( 2 \sin \sqrt{x} + c \)
• C. \( \frac{1}{2} \sin \sqrt{x} + c \)
• D. \( 2 \cos \sqrt{x} + c \)

Hint:

For integrals involving square roots, use substitution to simplify the expression and make the integration easier.

Answer 1

The Correct Option is B

Step 1: Use substitution.
Let \( u = \sqrt{x} \), then \( du = \frac{1}{2\sqrt{x}} dx \), so the integral becomes: \[ \int \frac{\cos \sqrt{x}}{\sqrt{x}} \, dx = \int 2 \cos u \, du \]
Step 2: Integrate.
Now, integrate \( \int 2 \cos u \, du = 2 \sin u + c \). Substituting back \( u = \sqrt{x} \), we get: \[ 2 \sin \sqrt{x} + c \]
Step 3: Conclusion.
The correct answer is (B) \( 2 \sin \sqrt{x} + c \).