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      Question

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      A Ο€ Correct Answer Incorrect Answer
      B 1/2 Correct Answer Incorrect Answer
      C 1 Correct Answer Incorrect Answer

      Solution

      First, let's determine if the function f(x) is odd or even. A function is even if f(βˆ’x)=f(x). A function is odd if f(βˆ’x)=βˆ’f(x). Let's evaluate f(βˆ’x): f(βˆ’x)=(βˆ’x)2sin(βˆ’x) f(βˆ’x)=x2(βˆ’sinx) f(βˆ’x)=βˆ’x2sinx f(βˆ’x)=βˆ’f(x) Since f(βˆ’x)=βˆ’f(x), the functionf(x) = x2sin(x)is an odd function. For an odd function f(x), the definite integral over a symmetric interval [βˆ’a, a] is always zero: Therefore, the correct answer is option (D).

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