Let g t = ∫ - π / 2 π / 2 cos π 4 t + f x d x , where f x = log e x + x 2 + 1 , x ∈ R . Then which one of the following is correct?

Step-by-step solution

We have, f x = log e x + x 2 + 1 , x ∈ R ⇒ f x = log e x + x 2 + 1 x - x 2 + 1 x - x 2 + 1 ⇒ f x = log e - 1 x - x 2 + 1 ⇒ f x = log e 1 x 2 + 1 - x ⇒ f - x = log e 1 x 2 + 1 + x ⇒ f - x = log e x 2 + 1 + x - 1 ⇒ f - x = - log e x 2 + 1 + x ⇒ f - x = - f x Hence, f x is an odd function. Now, g t = ∫ - π / 2 π / 2 cos π 4 t + f x d x ⇒ g t = cos π 4 t ∫ - π / 2 π / 2 1   d x + ∫ - π / 2 π / 2 f x d x