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\(\int_{-a}^{a}f(x)dx=0,\) if :
UNDERSTAND KNOWLEDGE 1 Marks
Concept Application
50%
Calculation / Logic
50%
Target Level
MEDIUM
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UNDERSTAND KNOWLEDGE MEDIUM

Q: \(\int_{-a}^{a}f(x)dx=0,\) if :

Question Analysis & Solution

Step-by-Step Solution

  1. Recall the properties of definite integrals: Specifically, the property related to even and odd functions integrated over symmetric intervals.
  2. Consider the case when f(x) is an odd function: An odd function satisfies the condition \(f(-x) = -f(x)\). The integral of an odd function over a symmetric interval \([-a, a]\) is zero. That is, \(\int_{-a}^{a}f(x)dx = 0\) if \(f(-x) = -f(x)\).
  3. Consider the case when f(x) is an even function: An even function satisfies the condition \(f(-x) = f(x)\). The integral of an even function over a symmetric interval \([-a, a]\) is twice the integral from 0 to a. That is, \(\int_{-a}^{a}f(x)dx = 2\int_{0}^{a}f(x)dx\) if \(f(-x) = f(x)\). This is not equal to zero unless the integral from 0 to a is zero.
  4. Analyze the other options: Options (C) and (D) involve the property \(f(a-x)\), which relates to symmetry about the line \(x = a/2\), and doesn't directly imply that the integral from -a to a is zero.
  5. Conclusion: The correct condition for \(\int_{-a}^{a}f(x)dx = 0\) is when \(f(-x) = -f(x)\), which means f(x) is an odd function.

Correct Answer: f(-x)=-f(x)

AI Suggestion: Option B
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