Ex 7.3

Chapter 7 Class 12 Integrals
Serial order wise

### Transcript

Ex 7.3, 3 Integrate the function - cos 2x cos 4x cos 6x We know that 2 cos A cos B=[cosโก(๐ด+๐ต)+cosโก(๐ดโ๐ต) ] Replace A by 2๐ฅ & B by 4๐ฅ 2 cos 2๐ฅ cos 4๐ฅ=cosโก(2๐ฅ+4๐ฅ)+cosโก(2๐ฅโ4๐ฅ) 2(cos 2๐ฅ cos 4๐ฅ)=cosโกใ (6๐ฅ)ใ+cosโก(โ2๐ฅ) 2 cos 2๐ฅ cos 4๐ฅ=cosโก6๐ฅ+cosโก2๐ฅ cos 2๐ฅ cos 4๐ฅ=1/2 (cosโก6๐ฅ+cosโก2๐ฅ ) (โต๐๐๐ โก(โ๐ฅ)=๐๐๐ โก๐ฅ) Now, โซ1โ(cosโก2๐ฅ cosโก4๐ฅ cosโก6๐ฅ ) ๐๐ฅ =โซ1โ(1/2 (cosโก6๐ฅ+cosโก2๐ฅ ) cosโก6๐ฅ ) ๐๐ฅ =1/2 โซ1โใ(cosโก6๐ฅ+cosโก2๐ฅ ) cosโก6๐ฅ ใ ๐๐ฅ =1/2 โซ1โใ(cosโก6๐ฅ )^2+(cosโก2๐ฅ.cosโก6๐ฅ ) ใ ๐๐ฅ =1/2 [โซ1โใ(cosโก6๐ฅ )^2 ๐๐ฅ+ใ โซ1โใcosโก2๐ฅ.cosโก6๐ฅ ใ ๐๐ฅ] Solving both these integrals separately โซ1โ(ใ๐๐จ๐ฌใ^๐โก๐๐ ) We know that cos 2๐=2 cos^2โก๐โ1 cos^2 ๐=(cosโก2๐ + 1)/2 Replace ๐ by 6๐ฅ cos^2 6๐ฅ=cosโกใ2(6๐ฅ) + 1ใ/2 cos^2 6๐ฅ=cosโกใ12๐ฅ + 1ใ/2 โซ1โcos^2โกใ6๐ฅ ๐๐ฅใ =(โซ1โcosโกใ12๐ฅ + 1ใ )/2 ๐๐ฅ =1/2 โซ1โ(cosโกใ12๐ฅ+1ใ ) ๐๐ฅ โซ1โ๐๐จ๐ฌโก๐๐ ๐๐จ๐ฌ ๐๐ ๐๐ We know that 2 cos A cos B=cos (๐ด+๐ต)+cosโก(๐ดโ๐ต) cos A cos B=1/2 [cos (๐ด+๐ต)+cosโก(๐ดโ๐ต) ] Replace A by 2๐ฅ & B by 6๐ฅ cos 2๐ฅ cos 6๐ฅ =1/2 [cosโก(2๐ฅ+6๐ฅ)+cosโก(2๐ฅโ6๐ฅ) ] =1/2 [cosโก8๐ฅ+cosโก(โ4๐ฅ) ] =1/2 [cosโก8๐ฅ+cosโก4๐ฅ ] ๐๐ฅ โซ1โใ๐๐๐ โก2๐ฅ cosโก6๐ฅ ใ ๐๐ฅ =1/2 โซ1โ(cosโก8๐ฅ+cosโก4๐ฅ ) ๐๐ฅ Thus, our equation becomes โซ1โ(cosโก2๐ฅ cosโก4๐ฅ cosโก6๐ฅ ) ๐๐ฅ =1/2 [โซ1โcos^2โก6๐ฅ ๐๐ฅ+โซ1โ๐๐๐ โก2๐ฅ cosโก6๐ฅ ๐๐ฅ] =1/2 [1/2 โซ1โ(๐๐๐ โกใ12๐ฅ+1ใ ) ๐๐ฅ+1/2 โซ1โ(๐๐๐ โก8๐ฅ+cosโก4๐ฅ ) ๐๐ฅ] =1/4 [โซ1โ(๐๐๐ โกใ12๐ฅ+1ใ ) ๐๐ฅ+โซ1โใ(๐๐๐ โก8๐ฅ ใ+cosโก4๐ฅ)๐๐ฅ] =1/4 [โซ1โ๐๐๐ โก12๐ฅ ๐๐ฅ+โซ1โ1 ๐๐ฅ+โซ1โcosโก8๐ฅ ๐๐ฅ+โซ1โcosโก4๐ฅ ๐๐ฅ] =๐/๐ [๐ฌ๐ข๐งโก๐๐๐/๐๐ +๐+๐๐๐โก๐๐/๐ + ๐๐๐โก๐๐/๐]+๐ช โซ1โcosโก(๐๐ฅ+๐) ๐๐ฅ=๐ ๐๐โก(๐๐ฅ + ๐)/๐ =1/4 [โซ1โ(๐๐๐ โกใ12๐ฅ+1ใ ) ๐๐ฅ+โซ1โใ(๐๐๐ โก8๐ฅ ใ+cosโก4๐ฅ)๐๐ฅ] =1/4 [โซ1โ๐๐๐ โก12๐ฅ ๐๐ฅ+โซ1โ1 ๐๐ฅ+โซ1โcosโก8๐ฅ ๐๐ฅ+โซ1โcosโก4๐ฅ ๐๐ฅ] =๐/๐ [๐ฌ๐ข๐งโก๐๐๐/๐๐ +๐+๐๐๐โก๐๐/๐ + ๐๐๐โก๐๐/๐]+๐ช