Chapter 7 Class 12 Integrals

Chapter 8 Class 12 Application of Integrals →
Class 12
Important Questions for exams Class 12

Example 28 - Chapter 7 Class 12 Integrals (Important Question)

Last updated at July 14, 2026 by

  Example 28 - Evaluate integral -1 -> 2 |x3 - x| dx - Examples - Examples

part 2 - Example 28 - Examples - Serial order wise - Chapter 7 Class 12 Integrals
part 3 - Example 28 - Examples - Serial order wise - Chapter 7 Class 12 Integrals

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Transcript

Example 28 Evaluate āˆ«_(āˆ’1)^2ā–’|š‘„^3āˆ’š‘„| š‘‘š‘„ |š‘„^3āˆ’š‘„|=|š‘„(š‘„^2āˆ’1)| =|š‘„| |š‘„^2āˆ’1| =|š‘„| |(š‘„āˆ’1)(š‘„+1)| =|š‘„| |š‘„āˆ’1||š‘„+1| Thus, š‘„=0, š‘„=1 & š‘„=āˆ’1 āˆ“ |š‘„^3āˆ’š‘„|= {ā–ˆ((āˆ’š‘„)(āˆ’(š‘„āˆ’1))(āˆ’(š‘„+1)) š‘–š‘“ š‘„<āˆ’1@(āˆ’š‘„)"(āˆ’(š‘„āˆ’1))" (š‘„+1) š‘–š‘“ āˆ’1ā‰¤š‘„<0@(š‘„)"(āˆ’(š‘„āˆ’1))" (š‘„+1) š‘–š‘“ 0ā‰¤š‘„<1@(š‘„)(š‘„āˆ’1)(š‘„+1) š‘–š‘“ š‘„ā‰„1)ā”¤ |š‘„^3āˆ’š‘„|= {ā–ˆ(āˆ’š‘„(š‘„āˆ’1)(š‘„+1) š‘–š‘“ š‘„<āˆ’1@š‘„(š‘„āˆ’1)(š‘„+1) š‘–š‘“ āˆ’1ā‰¤š‘„<0@āˆ’š‘„(š‘„āˆ’1)(š‘„+1) š‘–š‘“ 0ā‰¤š‘„<1@š‘„(š‘„āˆ’1)(š‘„+1) š‘–š‘“ š‘„ā‰„1)ā”¤ |š‘„^3āˆ’š‘„|= {ā–ˆ(āˆ’(š‘„^3āˆ’š‘„) š‘–š‘“ š‘„<āˆ’1@(š‘„^3āˆ’š‘„) š‘–š‘“ āˆ’1ā‰¤š‘„<0@āˆ’(š‘„^3āˆ’š‘„) š‘–š‘“ 0ā‰¤š‘„<1@(š‘„^3āˆ’š‘„) š‘–š‘“ š‘„ā‰„1)ā”¤ Thus, our integration becomes āˆ«_(āˆ’1)^2ā–’|š‘„^3āˆ’š‘„| š‘‘š‘„ =āˆ«_(āˆ’1)^0ā–’(š‘„^3āˆ’š‘„) š‘‘š‘„āˆ’āˆ«_0^1ā–’(š‘„^3āˆ’š‘„) š‘‘š‘„+āˆ«_1^2ā–’(š‘„^3āˆ’š‘„) š‘‘š‘„ =āˆ«_(āˆ’1)^0ā–’š‘„^3 š‘‘š‘„āˆ’āˆ«_(āˆ’1)^0ā–’š‘„ š‘‘š‘„āˆ’āˆ«_0^1ā–’š‘„^3 š‘‘š‘„+āˆ«_0^1ā–’š‘„ š‘‘š‘„+āˆ«_1^2ā–’š‘„^3 š‘‘š‘„āˆ’āˆ«_1^2ā–’š‘„ š‘‘š‘„ =[š‘„^4/4]_(āˆ’1)^0āˆ’[š‘„^2/2]_(āˆ’1)^0āˆ’[š‘„^4/4]_0^1+[š‘„^2/2]_0^1+[š‘„^4/4]_1^2āˆ’[š‘„^2/2]_1^2 =[(0 āˆ’ (āˆ’1)^4)/4]āˆ’[(0 āˆ’ (āˆ’1)^2)/2]āˆ’[((1)^4 āˆ’ 0)/4]+[((1)^2 āˆ’ 0)/2]+[((2)^4 āˆ’ (1)^4)/4]āˆ’[((2)^2 āˆ’ (1)^2)/2] =(āˆ’1)/4 āˆ’ [(āˆ’1)/2] āˆ’ 1/4 + 1/2 + [(16 āˆ’ 1)/4]āˆ’[(4 āˆ’ 1)/2] =(āˆ’1)/4 + 1/2 āˆ’ 1/4 + 1/2 +15/4 āˆ’ 3/2 =(āˆ’1 + 2 āˆ’ 1 + 2 + 15 āˆ’ 6)/4 =šŸšŸ/šŸ’

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