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

Question

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, ∫ 5 x + 5 g ( t ) d t, ∫ x + 5 5 g ( t ) d t, 5 ∫ x + 5 5 g ( t ) d t, 2 ∫ 5 x + 5 g ( t ) d t

MARKS App · Accepted Answer

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 ) | - 5 x - 5 = f ( x - 5 ) - f - 5 ⇒ I = f ( 5 ) - f 5 + x ( f is an odd function and g is given as an even function) ⇒ I = ∫ 5 + x 5 f ' ( t ) d t = ∫ 5 + x 5 g ( t ) d t

Search any question & find its solution

Looking for more such questions to practice?