Chapter 7 Class 12 Integrals
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Ex 7.10, 19 - Chapter 7 Class 12 Integrals

Last updated at April 16, 2024 by

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Ex 7.10, 19 Show that ∫_0^𝑎▒𝑓(𝑥) 𝑔 (𝑥) 𝑑𝑥=2∫_0^𝑎▒𝑓(𝑥) 𝑑𝑥, if f and g are defined as 𝑓(𝑥)=𝑓(𝑎−𝑥) and 𝑔(𝑥)+𝑔(𝑎−𝑥)=4 Let I =∫_0^𝑎▒𝑓(𝑥) 𝑔(𝑥) 𝑑𝑥 I =∫_0^𝑎▒𝑓(𝑥) [4−𝑔(𝑎−𝑥)] 𝑑𝑥 I = ∫_0^𝑎▒[4.𝑓(𝑥)−𝑓(𝑥)𝑔(𝑎−𝑥)] 𝑑𝑥 I = 4∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥−∫_0^𝑎▒〖𝑓(𝑥) 𝑔(𝑎−𝑥) 〗〗 𝑑𝑥 I = 4∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥−∫_0^𝑎▒〖𝑓(𝑎−𝑥) 𝑔(𝑎−(𝑎−𝑥)) 〗〗 𝑑𝑥 I = 4∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥−∫_0^𝑎▒〖𝑓(𝑥) 𝑔(𝑥) 〗〗 𝑑𝑥 I =4∫_0^𝑎▒〖𝑓(𝑥)𝑑𝑥−I〗 I +I=4∫_0^𝑎▒𝑓(𝑥)𝑑𝑥 2I=4∫_0^𝑎▒𝑓(𝑥)𝑑𝑥 I=2∫_0^𝑎▒𝑓(𝑥)𝑑𝑥 ∴ ∫_0^𝑎▒〖𝑓(𝑥) 𝑔(𝑥) 〗 𝑑𝑥=2∫_0^𝑎▒𝑓(𝑥)𝑑𝑥 Hence Proved

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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 and Computer Science at Teachoo