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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Ex 7.8, 16 āˆ«_1^2ā–’(5š‘„^2)/(š‘„^2 + 4š‘„ + 3) š‘‘š‘„ Let F(š‘„)=āˆ«1ā–’怖(5š‘„^2)/(š‘„^2 + 4š‘„ + 3) š‘‘š‘„怗 =5āˆ«1ā–’怖š’™^šŸ/(š‘„^2 + 4š‘„ + 3) š‘‘š‘„怗 =5āˆ«1ā–’怖(š’™^šŸ + šŸ’š’™ + šŸ‘ āˆ’ šŸ’š’™ āˆ’ šŸ‘)/(š’™^šŸ + šŸ’š’™ + šŸ‘) š‘‘š‘„怗 =5āˆ«1ā–’怖(š‘„^2 + 4š‘„ + 3)/(š‘„^2 + 4š‘„ + 3) š‘‘š‘„怗āˆ’5āˆ«1ā–’怖( (4š‘„ + 3))/(š‘„^2 + 4š‘„ + 3) š‘‘š‘„怗 =āˆ«1ā–’怖(šŸ“āˆ’(šŸšŸŽš’™ + šŸšŸ“ )/(š’™^šŸ + šŸ’š’™ + šŸ‘)) š’…š’™ć€— =āˆ«1ā–’怖(5āˆ’(20š‘„ + 15 )/(š‘„^2 + 3š‘„ + š‘„ + 3)) š‘‘š‘„怗 =āˆ«1ā–’怖(5āˆ’(20š‘„ + 15 )/(š‘„(š‘„ + 3) + 1(š‘„ + 3))) š‘‘š‘„怗 =āˆ«1ā–’怖(šŸ“āˆ’(šŸšŸŽš’™ + šŸšŸ“ )/((š’™ + šŸ‘) (š’™ + šŸ))) š’…š’™ć€— Now, Let (šŸšŸŽš’™ + šŸšŸ“)/(š’™ + šŸ‘)(š’™ + šŸ) =š€/(š’™ + šŸ‘)+š/(š’™ + šŸ) (20š‘„ + 15)/(š‘„ + 3)(š‘„ + 1) =(A(š‘„ + 1) + B(š‘„ + 3))/(š‘„ + 3)(š‘„ + 1) Canceling denominator 20š‘„+15=A(š‘„ + 1) + B(š‘„ + 3) Putting š’™=āˆ’šŸ 20(āˆ’1)+15=A(āˆ’1 + 1) + B(āˆ’1 + 3) āˆ’20+15=AƗ0+B (2) āˆ’5=2B š=(āˆ’šŸ“)/( šŸ) Putting š’™=āˆ’šŸ‘ 20(āˆ’3)+15=A(āˆ’3+1)+B(āˆ’3+3) āˆ’60+15=A(āˆ’2) BƗ0 āˆ’45=āˆ’2A š‘Ø=šŸ’šŸ“/šŸ Hence āˆ«1ā–’ā–ˆ((šŸ“š’™^šŸ)/(š’™^šŸ + šŸ’š’™ + šŸ‘) " " =āˆ«1ā–’怖(5āˆ’(20š‘„ + 15 )/(š‘„^2 + 4š‘„ + 3)) š‘‘š‘„怗) =āˆ«1ā–’怖5āˆ’A/(š‘„ + 3)āˆ’怗 B/(š‘„ + 1) š‘‘š‘„ =āˆ«1ā–’怖šŸ“ š’…š’™ć€—āˆ’āˆ«1ā–’怖(šŸ’šŸ“/šŸ)/(š’™ + šŸ‘) š’…š’™āˆ’āˆ«1ā–’怖(((āˆ’šŸ“)/( šŸ)))/(š’™ + šŸ) š’…š’™ć€—怗 =5š‘„āˆ’45/2 š‘™š‘œš‘”|š‘„+3|+5/2 š‘™š‘œš‘”|š‘„+1| Hence F(š’™)=šŸ“š’™āˆ’šŸ“/šŸ [šŸ— š’š’š’ˆ|š’™+šŸ‘|āˆ’š’š’š’ˆ|š’™+šŸ|] Now, āˆ«_1^2ā–’怖(šŸ“š’™^šŸ)/(š’™^šŸ + šŸ’š’™ + šŸ‘) š’…š’™=š¹(2)āˆ’š¹(1) 怗 =[5 Ɨ 2āˆ’5/2 (9 š‘™š‘œš‘”|2+3|āˆ’š‘™š‘œš‘”|2+1|)] āˆ’ [5 Ɨ 1āˆ’5/2 (9 š‘™š‘œš‘”|1+3|āˆ’š‘™š‘œš‘”|1+1|)] =10āˆ’5/2 [9 š‘™š‘œš‘” 5āˆ’š‘™š‘œš‘” 3]āˆ’5+5/2 [9 š‘™š‘œš‘” 4āˆ’š‘™š‘œš‘” 2] =10āˆ’5āˆ’5/2 [9š‘™š‘œš‘” 5āˆ’š‘™š‘œš‘” 3āˆ’9š‘™š‘œš‘” 4+š‘™š‘œš‘” 2)] =10āˆ’5āˆ’5/2 [9š‘™š‘œš‘” 5āˆ’9š‘™š‘œš‘” 4āˆ’(š‘™š‘œš‘” 3āˆ’š‘™š‘œš‘” 2)] =šŸ“āˆ’šŸ“/šŸ (šŸ— š„šØš  šŸ“/šŸ’āˆ’š„šØš  šŸ‘/šŸ)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.