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Ex 7.8

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Ex 7.8, 3

Ex 7.8, 4 Important

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Ex 7.8, 15 You are here

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Ex 7.8, 21 (MCQ) Important

Ex 7.8, 22 (MCQ)

Last updated at June 13, 2023 by Teachoo

Ex 7.8, 15 โซ_0^1โใ๐ฅ๐^(๐ฅ^2 ) ใ ๐๐ฅ Step 1 :- Let F(๐ฅ)=โซ1โใ๐ฅ๐^(๐ฅ^2 ) ๐๐ฅใ Let ๐ฅ^2=๐ก Differentiating w.r.t. ๐ฅ 2๐ฅ=๐๐ก/๐๐ฅ ๐๐ฅ=๐๐ก/2๐ฅ Therefore, โซ1โใ๐ฅ ๐^(๐ฅ^2 ) ๐๐ฅ=โซ1โใ๐ฅ ๐^๐ก ๐๐ก/2๐ฅใใ =1/2 โซ1โใ๐^๐ก ๐๐กใ =1/2 ๐^๐ก Putting ๐ก=๐ฅ^2 =1/2 ๐^(๐ฅ^2 ) Hence, F(๐ฅ)=1/2 ๐^(๐ฅ^2 ) Step 2 :- โซ_0^1โใ๐ฅ๐^(๐ฅ^2 ) ๐๐ฅใ=๐น(1)โ๐น(0) =1/2 ๐^(1^2 )โ1/2 ๐^(0^2 ) =1/2 ๐^1โ1/2 ๐^0 =๐/๐ (๐โ๐)