Chapter 7 Class 12 Integrals
Concept wise

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

 

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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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo