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NCERT Solutions for Class 12 Help.html Mathematics Part Ii Chapter 7 Integrals

Integrals Here is the CBSE Help.html Chapter 7 for Class 12 students. Summary and detailed explanation of the lesson, including the definitions of difficult words. All of the exercises and questions and answers from the lesson's back end have been completed. NCERT Solutions for Class 12 Help.html Integrals Chapter 7 NCERT Solutions for Class 12 Help.html Integrals Chapter 7 The following is a summary in Hindi and English for the academic year 2025-26. You can save these solutions to your computer or use the Class 12 Help.html.

Question 1
CBSEENMA12032343

Evaluate integral from 1 to 2 of straight x space dx as the limit of a sum.

Solution

Comparing integral from 1 to 2 of straight x space dx space space with space integral from straight a to straight b of straight f left parenthesis straight x right parenthesis space dx comma space space we space get comma
               straight f left parenthesis straight x right parenthesis space equals space straight x comma space space space straight a space equals space 1 comma space space straight b space equals space 2
therefore space space space straight f left parenthesis straight a right parenthesis space equals space straight a comma space space straight f left parenthesis straight a plus straight h right parenthesis space equals space straight a plus straight h comma space space straight f left parenthesis straight a plus 2 straight h right parenthesis space equals space straight a plus 2 straight h comma space...... straight f left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis space equals space straight a plus left parenthesis straight n minus 1 right parenthesis straight h
Now   integral from straight a to straight b of straight f left parenthesis straight x right parenthesis space equals space stack Lt. straight h with straight h rightwards arrow 0 below left square bracket space straight f left parenthesis straight a right parenthesis plus straight f left parenthesis straight a plus straight h right parenthesis plus straight f left parenthesis straight a plus 2 straight h right parenthesis space plus space....... plus straight f left parenthesis straight a plus stack straight n minus 1 with bar on top space straight h right parenthesis right square bracket
                                                                                              where n h = b - a
rightwards double arrow        integral from straight a to straight b of xdx space equals space Lt with straight h rightwards arrow 0 below straight h left square bracket straight a plus straight a left parenthesis straight a plus straight h right parenthesis plus left parenthesis straight a plus 2 space straight h right parenthesis space plus space... plus left parenthesis straight a plus stack straight n minus 1 with bar on top straight h right parenthesis right square bracket
                          equals stack Lt space straight h with straight h rightwards arrow 0 below left square bracket left parenthesis straight a plus straight a plus straight a plus... to space straight n space terms right parenthesis space plus space straight h open curly brackets 1 plus 2 plus 3 plus.... plus left parenthesis straight n minus 1 right parenthesis close curly brackets right square bracket
                         equals space Lt with straight h rightwards arrow 0 below straight h open square brackets na plus straight h fraction numerator left parenthesis straight n minus 1 right parenthesis space left parenthesis straight n minus 1 plus 1 right parenthesis over denominator 2 end fraction close square brackets space space space space space open square brackets therefore 1 plus 2 plus 3 plus... plus straight n space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets
equals Lt with straight h rightwards arrow 0 below straight h open square brackets straight n space straight a plus fraction numerator straight n left parenthesis straight n minus 1 right parenthesis over denominator 2 end fraction straight h close square brackets space space equals Lt with straight h rightwards arrow 0 below open square brackets straight a space left parenthesis nh right parenthesis space plus space fraction numerator straight n space straight h left parenthesis nh minus straight h right parenthesis over denominator 2 end fraction close square brackets
equals Lt with straight h rightwards arrow 0 below open square brackets straight a space left parenthesis straight b minus straight a right parenthesis space plus space fraction numerator left parenthesis straight b minus straight a right parenthesis space left parenthesis straight b minus straight a minus straight h right parenthesis over denominator 2 end fraction close square brackets
equals space straight a left parenthesis straight b minus straight a right parenthesis space plus space fraction numerator left parenthesis straight b minus straight a right parenthesis space left parenthesis straight b minus straight a minus 0 right parenthesis over denominator 2 end fraction space equals space straight a space left parenthesis straight b minus straight a right parenthesis space plus fraction numerator left parenthesis straight b minus straight a right parenthesis squared over denominator 2 end fraction
equals space left parenthesis straight b minus straight a right parenthesis open square brackets straight a plus fraction numerator straight b minus straight a over denominator 2 end fraction close square brackets space equals space left parenthesis straight b minus straight a right parenthesis open parentheses fraction numerator straight b plus straight a over denominator 2 end fraction close parentheses
therefore space space space space space integral from straight a to straight b of xdx space equals space 1 half left parenthesis straight b squared minus straight a squared right parenthesis
Put a = 1,  b = 2
therefore space space space space space space space space space space space space space integral from 1 to 2 of xdx space equals space 1 half left parenthesis 4 minus 1 right parenthesis space equals space 3 over 2

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