Integration Full Chapter Explained - Integration Class 12 - Everything you need




Last updated at Dec. 20, 2019 by Teachoo
Transcript
Ex 7.4, 15 Integrate the function 1/โ((๐ฅ โ ๐)(๐ฅ โ ๐)) โซ1โ1/โ((๐ฅ โ ๐) (๐ฅ โ ๐)) ๐๐ฅ =โซ1โ1/โ(๐ฅ(๐ฅ โ ๐) โ ๐(๐ฅ โ ๐)) ๐๐ฅ =โซ1โ1/โ(๐ฅ^2 โ ๐๐ฅ โ ๐๐ฅ + ๐๐) ๐๐ฅ =โซ1โ1/โ(๐ฅ^2 โ ๐ฅ(๐ + ๐) + ๐๐) ๐๐ฅ =โซ1โ1/โ(๐ฅ^2 โ 2(๐ฅ)((๐ + ๐)/2) + ๐๐) ๐๐ฅ =โซ1โ1/โ(๐ฅ^2 โ 2(๐ฅ)((๐ + ๐)/2) + ((๐ + ๐)/2)^2โ ((๐ + ๐)/2)^2+ ๐๐) ๐๐ฅ =โซ1โ1/โ((๐ฅ โ (๐ + ๐)/2)^2 โ ((๐ + ๐)/2)^2+ ๐๐) ๐๐ฅ =โซ1โ1/โ((๐ฅ โ (๐ + ๐)/2)^2 โ ((๐^2 + ๐^2+ 2๐๐)/4) + ๐๐) ๐๐ฅ =โซ1โ1/โ((๐ฅ โ (๐ + ๐)/2)^2+ (โ ๐^2 โ ๐^2 โ 2๐๐ + 4๐๐)/4) ๐๐ฅ =โซ1โ1/โ((๐ฅ โ (๐ + ๐)/2)^2 + (โ ๐^2 โ ๐^2 + 2๐๐)/4) ๐๐ฅ =โซ1โ1/(โ((๐ฅ โ (๐ + ๐)/2)^2 โ ((๐^2 + ๐^2 โ 2๐๐)/4) ) ) ๐๐ฅ =โซ1โ1/(โ((๐ฅ โ (๐ + ๐)/2)^2 โ ((๐ โ ๐)/2)^2 ) ) ๐๐ฅ =๐๐๐โก|๐ฅ โ (๐ + ๐)/2 +โ((๐ฅ โ (๐ + ๐)/2)^2โ ((๐ โ ๐)/2)^2 )|+๐ถ =๐๐๐โก|๐ฅโ (๐ + ๐)/2 +โ(๐ฅ^2+((๐ + ๐)/2)^2โ2(๐ฅ)((๐ + ๐)/2)โ((๐ โ ๐)/2)^2 )|+๐ถ It is of form โซ1โ๐๐ฅ/โ(๐ฅ^2 โ ๐^2 ) =๐๐๐โก|๐ฅ+โ(๐ฅ^2โ๐^2 )|+๐ถ โด Replacing ๐ฅ by (๐ฅโ (๐ + ๐)/2) and a by ((๐ โ ๐)/2) , we get =๐๐๐โก|๐ฅโ (๐ + ๐)/2 +โ(๐ฅ^2โ2(๐ฅ)((๐ + ๐)/2)+((๐ + ๐)/2)^2โ((๐ โ ๐)/2)^2 )|+๐ถ =๐๐๐โก|๐ฅโ (๐ + ๐)/2 +โ(๐ฅ^2โ๐ฅ(๐+๐)+(๐^2 + ๐^2 + 2๐๐)/4โ(๐^2 + ๐^2 โ 2๐๐)/4)|+๐ถ =๐๐๐โก|๐ฅโ (๐ + ๐)/2 +โ(๐ฅ^2โ๐ฅ(๐+๐)+2๐๐/4+2๐๐/4) |+๐ถ =๐๐๐โก|๐ฅโ (๐ + ๐)/2 +โ(๐ฅ^2โ๐ฅ(๐+๐)+4๐๐/4) |+๐ถ =๐๐๐โก|๐ฅโ (๐ + ๐)/2 +โ(๐ฅ^2โ๐ฅ(๐+๐)+๐๐) |+๐ถ =๐๐๐โก|๐ฅโ (๐ + ๐)/2 +โ(๐ฅ^2โ๐๐ฅโ๐๐ฅ+๐๐) |+๐ถ =๐๐๐โก|๐ฅโ (๐ + ๐)/2 +โ(๐ฅ(๐ฅโ๐)โ๐(๐ฅโ๐) ) |+๐ถ =๐๐๐โก|๐โ (๐ + ๐)/๐ +โ((๐โ๐)(๐โ๐) ) |+๐ช
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