Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
Definite Integration by properties - P3
Definite Integration by properties - P3
Last updated at June 13, 2023 by Teachoo
Misc 40 Choose the correct answer If 𝑓(𝑎+𝑏−𝑥)=𝑓(𝑥), then ∫_𝑎^𝑏▒〖𝑥 𝑓(𝑥) 〗 𝑑𝑥 is equal to (A) (𝑎+𝑏)/2 ∫_𝑎^𝑏▒〖 𝑓(𝑏−𝑥) 〗 𝑑𝑥 (B) (𝑎+𝑏)/2 ∫_𝑎^𝑏▒〖 𝑓(𝑏+𝑥) 〗 𝑑𝑥 (C) (𝑏 −𝑎)/2 ∫_𝑎^𝑏▒〖 𝑓(𝑥) 〗 𝑑𝑥 (D) " " (𝑎+𝑏)/2 ∫_𝑎^𝑏▒〖 𝑓(𝑥) 〗 𝑑𝑥 Let I=∫_𝑎^𝑏▒〖𝑥 𝑓(𝑥) 〗 𝑑𝑥 ∴ I=∫_𝑎^𝑏▒〖(𝑎+𝑏−𝑥) 𝑓(𝑎+𝑏−𝑥) 〗 𝑑𝑥 I=∫_𝑎^𝑏▒〖(𝑎+𝑏−𝑥) 𝑓(𝑥) 〗 𝑑𝑥 I=∫_𝑎^𝑏▒〖(𝑎+𝑏) 𝑓(𝑥) 〗 𝑑𝑥−∫_𝑎^𝑏▒〖𝑥 𝑓(𝑥) 〗 𝑑𝑥 Adding (1) and (2) i.e (1) + (2) I+I=∫_𝑎^𝑏▒〖𝑥 𝑓(𝑥) 〗 𝑑𝑥+∫_𝑎^𝑏▒〖(𝑎+𝑏) 𝑓(𝑥) 〗 𝑑𝑥−∫_𝑎^𝑏▒〖𝑥 𝑓(𝑥) 〗 𝑑𝑥 2I=∫_𝑎^𝑏▒〖(𝑎+𝑏) 𝑓(𝑥) 〗 𝑑𝑥 2I=(𝑎+𝑏) ∫_𝑎^𝑏▒𝑓(𝑥) 𝑑𝑥 ∴ I=(𝑎 + 𝑏)/2 ∫_𝑎^𝑏▒𝑓(𝑥) 𝑑𝑥 ∴ Option D is correct .