Question

If \(\int _a^b(x^2-x)\,dx = 18,\int _a^bx^3\,dx = 0\), then \(a+b\) is equal to

• A. \(3\)
• B. \(6\)
• C. \(0\)
• D. \(9\)

Hint:

Use the odd function property of f(x) sin x over a symmetric interval.

Answer 1

The Correct Option is C

Step 1: Split the integrand:
\(\int_{-2}^2(|x| + f(x)\sin x)\,dx = \int_{-2}^2|x|\,dx + \int_{-2}^2f(x)\sin x\,dx\).

Step 2: Second integral:
Since f is even and \(\sin x\) is odd, \(f(x)\sin x\) is an odd function. The integral of an odd function over \([-2, 2]\) is 0.

Step 3: First integral:
\(|x|\) is even, so \(\int_{-2}^2|x|\,dx = 2\int_0^2x\,dx = 2\cdot\frac{4}{2} = 4\).
Total = \(4 + 0 = 4\).

Final Answer:
The value is 4, option (B). \[ \boxed{4} \]