Question

\(\int \sin^2 x \,dx =\)

• A. \(\frac{x}{2} + \frac{\sin 2x}{4} + c\)
• B. \(\frac{x}{2} - \frac{\cos 2x}{4} + c\)
• C. \(\frac{x}{2} + \frac{\cos 2x}{4} + c\)
• D. \(\frac{x}{2} - \frac{\sin 2x}{4} + c\)

Hint:

Memorize the power-reduction formulas for \(\sin^2 x\) and \(\cos^2 x\), as they are fundamental for integrating even powers of sine and cosine.

• \(\sin^2 x = \frac{1 - \cos(2x)}{2}\)

• \(\cos^2 x = \frac{1 + \cos(2x)}{2}\)
These are direct applications of the \(\cos(2x)\) double-angle identities.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Find ∫dx/(25-x²).

Step 2: Key Formula (Alternate):
Use standard formula: ∫dx/(a²-x²) = (1/2a)ln|(a+x)/(a-x)| + c.

Step 3: Detailed Explanation:
Here a=5. Result = (1/10)ln|(5+x)/(5-x)| + c. Can also be done by partial fractions.

Step 4: Final Answer:
Integral is (1/10)log|(5+x)/(5-x)| + c.