NTA Abhyas JEE Main2020MathematicsDefinite IntegrationPractice
Which of the following statements is correct?
Options
- AIf f x is an odd function, then g x = ∫ a x f t d t is an even function.
- BIf f x is an odd function, then g x = ∫ a x f t d t is also odd.
- CIf f x is a periodic function, then g x = ∫ a x f t d t is also periodic.
- DIf f x is periodic, then g x = ∫ a x f t d t is always non - periodic.
Correct answer
A. If f x is an odd function, then g x = ∫ a x f t d t is an even function.
Step-by-step solution
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.