Ex 7.6, 18 - Integrate ex (1 + sin x / 1 + cos x) - Ex 7.6

Slide10.JPG
Slide11.JPG Slide12.JPG

  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
Ask Download

Transcript

Ex7.6, 18 𝑒𝑥 1 + sin﷮𝑥﷯﷮1 + cos﷮𝑥﷯﷯﷯ Simplifying the given function 𝑒﷮𝑥﷯ 1 + sin﷮𝑥﷯﷮1 + cos﷮𝑥﷯﷯﷯ 𝑒﷮𝑥﷯ 1 + sin﷮𝑥﷯﷮1 + cos﷮𝑥﷯﷯﷯= 𝑒﷮𝑥﷯ 1 + 2 sin﷮ 𝑥﷮2﷯﷯﷯ cos﷮ 𝑥﷮2﷯﷯﷯﷮2 𝑐𝑜 𝑠﷮2﷯﷮ 𝑥﷮2﷯﷯﷯﷯﷯ 𝑒﷮𝑥﷯ 1 + sin﷮𝑥﷯﷮1 + cos﷮𝑥﷯﷯﷯= 𝑒﷮𝑥﷯ sin﷮2﷯﷮ 𝑥﷮2﷯﷯﷯ + cos﷮2﷯﷮ 𝑥﷮2﷯﷯﷯ +2 sin ﷮ 𝑥﷮2﷯﷯ . cos ﷮ 𝑥﷮2﷯﷯﷮ 2 𝑐𝑜 𝑠﷮2﷯﷮ 𝑥﷮2﷯﷯﷯ ﷯﷯ = 𝑒﷮𝑥﷯ sin ﷮ 𝑥﷮2﷯﷯ + cos ﷮ 𝑥﷮2﷯﷯﷯﷮2﷯﷮ 2 𝑐𝑜 𝑠﷮2﷯﷮ 𝑥﷮2﷯﷯﷯ ﷯﷯ = 𝑒﷮𝑥﷯﷮2﷯ sin ﷮ 𝑥﷮2﷯﷯ + cos ﷮ 𝑥﷮2﷯﷯﷮ 𝑐𝑜𝑠﷮ 𝑥﷮2﷯﷯﷯﷯﷮2﷯ = 𝑒﷮𝑥﷯﷮2﷯ sin ﷮ 𝑥﷮2﷯﷯﷮ 𝑐𝑜𝑠﷮ 𝑥﷮2﷯﷯﷯ + cos ﷮ 𝑥﷮2﷯﷯﷮ 𝑐𝑜𝑠﷮ 𝑥﷮2﷯﷯﷯﷯﷮2﷯ = 𝑒﷮𝑥﷯﷮2﷯ tan ﷮ 𝑥﷮2﷯﷯ +1﷯﷮2﷯ = 𝑒﷮𝑥﷯﷮2﷯ tan﷮2﷯﷮ 𝑥﷮2﷯﷯ + 1﷯﷮2﷯+2 tan ﷮ 𝑥﷮2﷯﷯﷯ 1﷯﷯ = 𝑒﷮𝑥﷯﷮2﷯ tan﷮2﷯ ﷮ 𝑥﷮2﷯﷯ +1+2 tan ﷮ 𝑥﷮2﷯﷯﷯ = 𝑒﷮𝑥﷯﷮2﷯ sec﷮2﷯ ﷮ 𝑥﷮2﷯﷯ +2 tan ﷮ 𝑥﷮2﷯﷯﷯﷯ = 𝑒﷮𝑥﷯ 1﷮2﷯ . sec﷮2﷯﷮ 𝑥﷮2﷯﷯ + 2﷮2﷯ tan ﷮ 𝑥﷮2﷯﷯﷯﷯ = 𝑒﷮𝑥﷯ sec﷮2﷯﷮ 𝑥﷮2﷯﷯﷮2﷯ + tan ﷮ 𝑥﷮2﷯﷯﷯ Now, Integrating the function 𝑤.𝑟.𝑡.𝑥 ﷮﷮ 𝑒﷮𝑥﷯ 1 + sin﷮𝑥﷯﷮1 + cos﷮𝑥﷯﷯﷯ 𝑑𝑥﷯= ﷮﷮ 𝑒﷮𝑥﷯ sec﷮2﷯﷮ 𝑥﷮2﷯﷯﷮2﷯ + tan ﷮ 𝑥﷮2﷯﷯﷯ 𝑑𝑥﷯ Putting, 𝑓 𝑥﷯= tan ﷮ 𝑥﷮2﷯﷯ ∴ 𝑓﷮′﷯ 𝑥﷯= sec﷮2﷯﷮ 𝑥﷮2﷯﷯ 1﷮2﷯﷯= sec﷮2﷯﷮ 𝑥﷮2﷯﷯﷮2﷯ Thus, ﷮﷮ 𝑒﷮𝑥﷯ sec﷮2﷯﷮ 𝑥﷮2﷯﷯﷮2﷯ + tan ﷮ 𝑥﷮2﷯﷯﷯ 𝑑𝑥﷯ = 𝒆﷮𝒙﷯ 𝒕𝒂𝒏 ﷮ 𝒙﷮𝟐﷯﷯+𝑪

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.