Question

\(\int_{0}^{\pi/2} \frac{\cos 2x}{\sin x + \cos x} \,dx =\)

• A. -1
• B. 0
• C. 1
• D. \(\frac{\pi}{2}\)

Hint:

When faced with a trigonometric fraction in an integral, always look for identities that can simplify the expression. The double angle formulas, especially \(\cos 2x\), are very versatile and have multiple forms (\(\cos^2x - \sin^2x\), \(2\cos^2x - 1\), \(1 - 2\sin^2x\)). Choosing the right form is key. Here, the difference of squares form was perfect for cancellation.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Find area bounded by y=4x², x-axis, x=0, x=1.

Step 2: Key Formula (Alternate):
Area = ∫ₐᵇ f(x)dx where f(x)≥0 on [a,b].

Step 3: Detailed Explanation:
A = ∫₀¹ 4x² dx = 4[x³/3]₀¹ = 4(1/3) = 4/3.

Step 4: Final Answer:
Area is 4/3 sq units.