Which of the following statements is correct?

Question

Which of the following statements is correct?, If f x is an odd function, then g x = ∫ a x f t d t is an even function., If f x is an odd function, then g x = ∫ a x f t d t is also odd., If f x is a periodic function, then g x = ∫ a x f t d t is also periodic., If f x is periodic, then g x = ∫ a x f t d t is always non - periodic.

MARKS App · Accepted Answer

As, g x = ∫ 0 x f t d t ⇒ g - x = ∫ 0 - x f t d t Put t = - u ⇒ d t = - d u ∴ g - x = - ∫ 0 x f - u d u ∫ 0 x f u d u (as f x is odd) ∴ g - x = g x Hence, g x is even. Also, Consider g x = ∫ 0 x cos 2 ⁡ t d t ⇒ g x = x 2 + sin ⁡ 2 x 4 Which is non - periodic even though y = cos 2 ⁡ x is periodic Also, if g x = ∫ 0 x cos ⁡ t d t Then, g x = sin ⁡ x , which is periodic.

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