Question 5
If 𝑓 "and" 𝑔 are continuous functions in [0, 1] satisfying
𝑓(𝑥)=𝑓(𝑎−𝑥) and 𝑔(𝑥)+𝑔 (𝑎−𝑥)=𝑎, then ∫1_0^𝑎▒〖𝑓(𝑥). 𝑔(𝑥)〗 𝑑𝑥 is equal to
𝑎/2 (B) 𝑎/2 ∫1_0^𝑎▒〖𝑓(𝑥) 𝑑𝑥〗
(C) ∫1_0^𝑎▒〖𝑓(𝑥) 𝑑𝑥〗 (D) 𝑎∫1_0^𝑎▒〖𝑓(𝑥) 𝑑𝑥〗
Let
𝑰 =∫_𝟎^𝒂▒𝒇(𝒙) 𝒈(𝒙) 𝒅𝒙
Using g(𝑥)+𝑔(𝑎−𝑥)=𝑎
I =∫_0^𝑎▒𝑓(𝑥) [𝑎−𝑔(𝑎−𝑥)] 𝑑𝑥
I = ∫_0^𝑎▒[𝑎.𝑓(𝑥)−𝑓(𝑥)𝑔(𝑎−𝑥)] 𝑑𝑥
𝑰 =𝒂∫_𝟎^𝒂▒〖𝒇(𝒙)𝒅𝒙−∫_𝟎^𝒂▒〖𝒇(𝒙) 𝒈(𝒂−𝒙) 〗〗 𝒅𝒙
I =𝑎∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥−∫_𝟎^𝒂▒〖𝒇(𝒂−𝒙) 𝒈(𝒂−(𝒂−𝒙)) 〗〗 𝑑𝑥
I =𝑎∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥−∫_0^𝑎▒〖𝒇(𝒂−𝒙) 𝑔(𝑥) 〗〗 𝑑𝑥
Using 𝑓(𝑥)=𝑓(𝑎−𝑥)
I =𝑎∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥−∫_0^𝑎▒〖𝒇(𝒙) 𝑔(𝑥) 〗〗 𝑑𝑥
I =𝑎∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥−𝐈〗
I +I=a∫_0^𝑎▒𝑓(𝑥)𝑑𝑥
2I=a∫_0^𝑎▒𝑓(𝑥)𝑑𝑥
𝐈=𝐚/𝟐 ∫_𝟎^𝒂▒𝒇(𝒙)𝒅𝒙
∴ ∫_0^𝑎▒〖𝑓(𝑥) 𝑔(𝑥) 〗 𝑑𝑥=2∫_0^𝑎▒𝑓(𝑥)𝑑𝑥
Hence Proved
So, the correct answer is (b)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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