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The Fourier series is given by   in the interval -π < x < π. 
Which among the following is true?
  • a)
    If f(x) = xcos⁡x, the value of a0 = 0
  • b)
    If f(x) = xcos⁡x, the value of an = 0
  • c)
    If f(x) = xsin⁡x, the value of an = 0
  • d)
    If f(x) = xsin⁡x, the value of bn = 0
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The Fourier series is given byin the interval -π < x < π.W...
Concept:
Fourier series:
The Fourier series for the function f(x) in the interval α < x < α + 2π is given by:

where

Even and odd function:

Calculation:
Given:

For f(x) = xcos⁡x
f(x) is an odd function
 [Option 1 is correct].

For f(x) = xcos⁡x is an odd function whereas cos⁡nx is an even function. 
The product of odd and even function is an odd function.
 [Option 2 is correct].
For f(x) = xsin⁡x is an even function whereas cos⁡nx is an even function.
The product of two even function is an even function.
 [Option 3 is incorrect].
For f(x) = xsin⁡x is an even function whereas sin⁡nx is an odd function.  
The product of even and odd function is an odd function.
 [Option 4 is correct].
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Question Description
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