Let f x = ∫ 0 x g t d t , where g is a non-zero even function. If f x + 5 = g x , then ∫ 0 x f ( t ) d t equals

Step-by-step solution

Given : g x is an even function, then f x = ∫ 0 x g ( t ) d t will be an odd function, ⇒ f - x = - f x Let I = ∫ 0 x f t d t put t = z + 5 ⇒ I = ∫ - 5 x - 5 f z + 5 d z ⇒ I = ∫ - 5 x - 5 g ( z ) d z     ∵ f x + 5 = g x ⇒ I = ∫ - 5 x - 5 f ' ( z ) d z                         ( ∵ f ' ( x ) = g ( x ) ) ⇒ I = f ( z