Let, the function F be defined as F x = ∫ 1 x e t t d t , x > 0 , then the value of the integral ∫ 1 x e t t + a d t , where a > 0 , is

Step-by-step solution

Given F x = ∫ 1 x e t t d t Let, I = ∫ 1 x e t t + a d t Let, t + a = y ⇒ d t = d y Also, t = 1  ⇒ y = 1 + a and t = x  ⇒ y = x + a ∴   I = ∫ 1 + a x + a e y - a y d y ⇒ I = e - a ∫ 1 + a x + a e y y d y Using ∫ a b f x d x = ∫ a b f t d t , we get I = e - a ∫ 1 + a x + a e t t d t ⇒ I = e - a ∫ 1 1 + a e t t d t + ∫ 1 + a x + a e t t d t - ∫ 1 1 + a e t t d t Using ∫ a b f x d x + ∫ b c f