Definite Integration by properties - P4

Chapter 7 Class 12 Integrals
Concept wise

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Example 42 Evaluate โซ_0^๐โ(๐ฅ ๐๐ฅ)/(๐^2 cos^2โกใ๐ฅ + ๐^2 ใ sin^2โก๐ฅ )Let I= โซ_0^๐โใ๐ฅ/(๐^2 ๐๐๐ ^2 ๐ฅ + ๐^2 ๐ ๐๐^2 ๐ฅ) ๐๐ฅใ โด I=โซ_0^๐โใ((๐ โ ๐ฅ))/(๐^2 ๐๐๐ ^2 (๐ โ ๐ฅ) + ๐^2 ๐ ๐๐^2 (๐ โ ๐ฅ) ) ๐๐ฅใ I=โซ_0^๐โใ(๐ โ ๐ฅ)/(๐^2 [๐๐๐ (๐ โ ๐ฅ)]^2 + ๐^2 [๐ ๐๐(๐ โ ๐ฅ)]^2 ) ๐๐ฅใ I=โซ_0^๐โใ(๐ โ ๐ฅ)/(๐^2 [โ ๐๐๐  ๐ฅ]^2 + ๐^2 [๐ ๐๐ ๐ฅ]^2 ) ๐๐ฅใ I=โซ_0^๐โใ(๐ โ ๐ฅ)/(๐^2 cos^2โก๐ฅ + ๐^2 sin^2โก๐ฅ ) ๐๐ฅใ Adding (1) and (2) i.e. (1) + (2) I+I=โซ_0^๐โใ๐ฅ/(๐^2 cos^2โก๐ฅ + ๐^2 sin^2โก๐ฅ ) ๐๐ฅใ+โซ1โ(๐ โ ๐ฅ)/(๐^2 cos^2โก๐ฅ + ๐^2 sin^2โก๐ฅ ) ๐๐ฅ 2I=โซ_0^๐โ(๐ฅ + ๐ โ ๐ฅ)/(๐^2 cos^2โก๐ฅ + ๐^2 sin^2โก๐ฅ ) ๐๐ฅ 2I=โซ_0^๐โ(๐ )/(๐^2 cos^2โก๐ฅ + ๐^2 sin^2โก๐ฅ ) ๐๐ฅ I=๐/2 โซ_0^๐โใ1/(๐^2 cos^2โก๐ฅ + ๐^2 sin^2โก๐ฅ ) ๐๐ฅใ Dividing numerator and denominator by ๐๐๐ ^2 ๐ฅ, we get I=๐/2 โซ_0^๐โใ(1/cos^2โก๐ฅ )/((๐^2 cos^2โกใ๐ฅ + ๐^2 sin^2โก๐ฅ ใ)/cos^2โก๐ฅ ) ๐๐ฅใ I=๐/2 โซ_0^๐โใ(๐ ๐๐^2 ๐ฅ)/((๐^2 cos^2โก๐ฅ)/cos^2โก๐ฅ + (๐^2 sin^2โก๐ฅ)/cos^2โก๐ฅ ) ๐๐ฅใ I=๐/2 โซ_0^๐โใ(๐ ๐๐^2 ๐ฅ)/(๐^2 + ๐^2 tan^2โก๐ฅ ) ๐๐ฅใ Let ๐(๐ฅ)=sec^2โก๐ฅ/(๐^2 + ๐^2 tan^2โก๐ฅ ) and a = ฯ Now, ๐(2๐โ๐ฅ)=sec^2โก(๐ โ ๐ฅ)/(๐^2 + ๐^2 tan^2โก(๐ โ ๐ฅ) ) ๐(2๐โ๐ฅ)=[โ๐ ๐๐ ๐ฅ]^2/(๐^2 + ๐^2 [โtanโก๐ฅ ]^2 ) ๐(2๐โ๐ฅ)=(๐ ๐๐^2 ๐ฅ)/(๐^2 + ๐^2 tan^2โก๐ฅ ) Therefore, ๐(๐ฅ)=๐(2๐โ๐ฅ) Therefore, I=๐/2 โซ_0^๐โใ(๐ ๐๐^2 ๐ฅ)/(๐^2 + ๐^2 tan^2โก๐ฅ ) ๐๐ฅใ =๐/2 ร 2 โซ_0^(๐/2)โใ(๐ ๐๐^2 ๐ฅ)/(๐^2 + ๐^2 tan^2โก๐ฅ ) ๐๐ฅใ =๐โซ_0^(๐/2)โใ(๐ ๐๐^2 ๐ฅ)/(๐^2 + ๐^2 tan^2โก๐ฅ ) ๐๐ฅใ Let ๐ tanโกใ๐ฅ=๐กใ Differentiating both sides w.r.t. ๐ฅ ๐ ๐ ๐๐^2 ๐ฅ ๐๐ฅ=๐๐ก ๐๐ก=๐๐ก/(๐^2 ๐ ๐๐^2 ๐ฅ) Putting the values of tan ๐ฅ and ๐๐ฅ , we get ๐ผ=๐โซ1_0^(๐/2)โใ(๐ ๐๐^2 ๐ฅ)/(๐^2 + ๐ก^2 ) . ๐๐ฅใ ๐ผ=๐ โซ1_0^โโใ(๐ ๐๐^2 ๐ฅ)/(๐^2 + ๐ก^2 ) .๐๐ก/(๐ ๐ ๐๐^2 ๐ฅ)ใ ๐ผ=๐/๐ โซ1_0^โโ๐๐ก/(๐^2 + ๐ก^2 ) ๐ผ= ใ๐/๐ [1/๐ tan^(โ1)โก(๐ก/๐) ]ใ_0^โ Putting limits, I=๐/๐ [1/๐ ใ๐ก๐๐ใ^(โ1) (โ/๐)โ1/๐ ใ๐ก๐๐ใ^(โ1) (0/๐)] I =๐/๐ [ใ1/๐ tan^(โ1)ใโกใ(โ)โ1/๐ tan^(โ1)โก(0) ใ ] I =๐/๐ (1/๐ (๐/2)โ0) I =๐^๐/๐๐๐