Ex 7.8

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Ex 7.8, 16 Important You are here

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Ex 7.8, 21 (MCQ) Important

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Last updated at April 16, 2024 by Teachoo

Ex 7.8, 16 ā«_1^2ā(5š„^2)/(š„^2 + 4š„ + 3) šš„ Let F(š„)=ā«1āć(5š„^2)/(š„^2 + 4š„ + 3) šš„ć =5ā«1āćš^š/(š„^2 + 4š„ + 3) šš„ć =5ā«1āć(š^š + šš + š ā šš ā š)/(š^š + šš + š) šš„ć =5ā«1āć(š„^2 + 4š„ + 3)/(š„^2 + 4š„ + 3) šš„ćā5ā«1āć( (4š„ + 3))/(š„^2 + 4š„ + 3) šš„ć =ā«1āć(šā(ššš + šš )/(š^š + šš + š)) š šć =ā«1āć(5ā(20š„ + 15 )/(š„^2 + 3š„ + š„ + 3)) šš„ć =ā«1āć(5ā(20š„ + 15 )/(š„(š„ + 3) + 1(š„ + 3))) šš„ć =ā«1āć(šā(ššš + šš )/((š + š) (š + š))) š šć Now, Let (ššš + šš)/(š + š)(š + š) =š/(š + š)+š/(š + š) (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=āš 20(ā1)+15=A(ā1 + 1) + B(ā1 + 3) ā20+15=AĆ0+B (2) ā5=2B š=(āš)/( š) Putting š=āš 20(ā3)+15=A(ā3+1)+B(ā3+3) ā60+15=A(ā2) BĆ0 ā45=ā2A šØ=šš/š Hence ā«1āā((šš^š)/(š^š + šš + š) " " =ā«1āć(5ā(20š„ + 15 )/(š„^2 + 4š„ + 3)) šš„ć) =ā«1āć5āA/(š„ + 3)āć B/(š„ + 1) šš„ =ā«1āćš š šćāā«1āć(šš/š)/(š + š) š šāā«1āć(((āš)/( š)))/(š + š) š šćć =5š„ā45/2 ššš|š„+3|+5/2 ššš|š„+1| Hence F(š)=ššāš/š [š ššš|š+š|āššš|š+š|] Now, ā«_1^2āć(šš^š)/(š^š + šš + š) š š=š¹(2)āš¹(1) ć =[5 Ć 2ā5/2 (9 ššš|2+3|āššš|2+1|)] ā [5 Ć 1ā5/2 (9 ššš|1+3|āššš|1+1|)] =10ā5/2 [9 ššš 5āššš 3]ā5+5/2 [9 ššš 4āššš 2] =10ā5ā5/2 [9ššš 5āššš 3ā9ššš 4+ššš 2)] =10ā5ā5/2 [9ššš 5ā9ššš 4ā(ššš 3āššš 2)] =šāš/š (š š„šØš š/šāš„šØš š/š)