Question

\(\int_{-a}^{a} |x| \,dx =\)

• A. a
• B. 2a
• C. 0
• D. \(a^2\)

Hint:

Recognizing whether a function is even or odd is a powerful shortcut for definite integrals over symmetric intervals like \([-a, a]\).

• If \(f(x)\) is even, \(\int_{-a}^{a} f(x) \,dx = 2 \int_{0}^{a} f(x) \,dx\).

• If \(f(x)\) is odd, \(\int_{-a}^{a} f(x) \,dx = 0\).
This can simplify the calculation significantly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Evaluate ∫₀^(π/2) (cos 2x)/(sin x+cos x) dx.

Step 2: Key Formula (Alternate):
Use identity cos 2x = cos²x-sin²x = (cos x-sin x)(cos x+sin x).

Step 3: Detailed Explanation:
Cancel (cos x+sin x). I = ∫₀^(π/2) (cos x-sin x)dx = [sin x+cos x]₀^(π/2) = (1+0)-(0+1) = 0.

Step 4: Final Answer:
Integral is 0.