Example 22 - Chapter 7 Class 12 Integrals

Last updated at July 28, 2026 by

Example 22 - Find (i) ex ( tan-1 x + 1 / 1 + x2) dx - Examples

Example 22 - Chapter 7 Class 12 Integrals - Part 2
Example 22 - Chapter 7 Class 12 Integrals - Part 3 Example 22 - Chapter 7 Class 12 Integrals - Part 4 Example 22 - Chapter 7 Class 12 Integrals - Part 5

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Example 22 Find (i) āˆ«1ā–’š‘’^š‘„ (tan^(āˆ’1)ā”š‘„+ 1/(1 + š‘„^2 )) š‘‘š‘„ āˆ«1ā–’怖š‘’^š‘„ (tan^(āˆ’1)ā”š‘„+1/(1 + š‘„^2 ))š‘‘š‘„怗 It is of the form āˆ«1ā–’怖š‘’^š‘„ [š‘“(š‘„)+š‘“^ā€² (š‘„)] 怗 š‘‘š‘„=š‘’^š‘„ š‘“(š‘„)+š¶ Where š‘“(š‘„)=tan^(āˆ’1)ā”š‘„ š‘“^ā€² (š‘„)= 1/(1 + š‘„^2 ) So, our equation becomes āˆ«1ā–’怖š‘’^š‘„ (tan^(āˆ’1)ā”š‘„+1/(1 + š‘„^2 ))š‘‘š‘„怗=š’†^š’™ 怖š­ššš§ć€—^(āˆ’šŸ)ā”怖š’™+š‘Ŗ怗 Example 22 Find (ii) āˆ«1ā–’((š‘„^2 + 1) š‘’^š‘„)/(š‘„ + 1)^2 š‘‘š‘„ āˆ«1ā–’怖(š‘„^2 + 1)/(š‘„ + 1)^2 .š‘’^š‘„ š‘‘š‘„怗 Adding and subtracting 1 in numerator =āˆ«1ā–’怖(š‘„^2+ 1 + 1 āˆ’ 1)/(š‘„ + 1)^2 .š‘’^š‘„ .š‘‘š‘„怗 =āˆ«1ā–’怖(š‘„^2 āˆ’ 1 + 1 + 1)/(š‘„ + 1)^2 .š‘’^š‘„ .š‘‘š‘„怗 =āˆ«1ā–’怖[(š‘„^2 āˆ’ 1)/(š‘„ + 1)^2 +2/(š‘„ + 1)^2 ] š‘’^š‘„ š‘‘š‘„怗 =āˆ«1ā–’怖š‘’^š‘„ [(š‘„^2 āˆ’ (1)^2)/(š‘„ + 1)^2 +2/(š‘„ + 1)^2 ]š‘‘š‘„怗 =āˆ«1ā–’怖š‘’^š‘„ [(š‘„ āˆ’ 1)(š‘„ + 1)/(š‘„ + 1)^2 +2/(š‘„ + 1)^2 ]š‘‘š‘„怗 =āˆ«1ā–’怖š‘’^š‘„ [(š‘„ āˆ’ 1)/(š‘„ + 1)+2/(š‘„ + 1)^2 ]š‘‘š‘„怗 It is of form āˆ«1ā–’怖š‘’^š‘„ [š‘“(š‘„)+š‘“^ā€² (š‘„)] 怗 š‘‘š‘„=š‘’^š‘„ š‘“(š‘„)+š¶ Where š‘“(š‘„)=(š‘„ āˆ’ 1)/(š‘„ + 1) š‘“^ā€² (š‘„)=š‘‘/š‘‘š‘„ [(š‘„ āˆ’ 1)/(š‘„ + 1)] š‘“^ā€² (š‘„)=(1.(š‘„ + 1) āˆ’1 (š‘„ āˆ’ 1))/(š‘„ + 1)^2 =(š‘„ + 1 āˆ’ š‘„ + 1)/(š‘„ + 1)^2 =2/(š‘„ + 1)^2 Thus, our equation becomes āˆ«1ā–’怖(š‘„^2 + 1)/(š‘„ + 1)^2 .š‘’^š‘„=āˆ«1ā–’怖š‘’^š‘„ [(š‘„ āˆ’ 1)/(š‘„ + 1)+2/(š‘„ + 1)^2 ]š‘‘š‘„怗怗 =š‘’^š‘„ [(š‘„ āˆ’ 1)/(š‘„ + 1)]+š¶ =(š’™ āˆ’ šŸ)/(š’™ + šŸ).š’†^š’™+š‘Ŗ

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