Example 7 - Find intgeral (i) cos2 x dx (ii) sin 2x cos 3x

Example 7 - Chapter 7 Class 12 Integrals - Part 2
Example 7 - Chapter 7 Class 12 Integrals - Part 3

 

  1. Chapter 7 Class 12 Integrals (Term 2)
  2. Serial order wise

Transcript

Example 7 Find (i) ∫1β–’cos^2⁑π‘₯ 𝑑π‘₯ ∫1β–’cos^2⁑π‘₯ 𝑑π‘₯ =∫1β–’((cos⁑2π‘₯ + 1)/2) 𝑑π‘₯ = 1/2 ∫1β–’(cos⁑2π‘₯+1) 𝑑π‘₯ = 1/2 [∫1β–’cos⁑2π‘₯ 𝑑π‘₯+∫1β–’1 𝑑π‘₯] We know cos⁑2π‘₯=2 cos^2⁑π‘₯βˆ’1 cos⁑2π‘₯+1=2 cos^2⁑π‘₯ (cos⁑2π‘₯ + 1)/2=cos^2⁑π‘₯ ∫1β–’π’„π’π’”β‘πŸπ’™ 𝒅𝒙 Let 2π‘₯=𝑑 2 =𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=1/2 𝑑𝑑 ∫1β–’cos⁑𝑑 . 1/2 𝑑𝑑 =1/2 (sin⁑𝑑+𝐢1) Putting value of 𝑑 = 2π‘₯ =1/2 sin⁑2π‘₯+𝐢1 ∫1β–’πŸ 𝒅𝒙 =∫1β–’π‘₯^0 𝑑π‘₯ =[π‘₯^(0 + 1)/(0 + 1)]+𝐢 =π‘₯+𝐢2 Thus, ∫1β–’cos^2⁑π‘₯ 𝑑π‘₯=1/2 [∫1β–’cos⁑2π‘₯ 𝑑π‘₯+∫1β–’1 𝑑π‘₯] =1/2 [1/2 sin⁑2π‘₯+𝐢1+π‘₯+𝐢2] =1/4 sin⁑2π‘₯+π‘₯/2+1/2(𝐢1+𝐢2) =𝒙/𝟐+𝟏/πŸ’ π¬π’π§β‘πŸπ’™+π‘ͺ ("From (1) and (2) " ) ("Let" 𝐢1+𝐢2=𝐢)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.