Example 7 (i) - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 7 Find (i) ∫1▒cos^2𝑥 𝑑𝑥 ∫1▒cos^2𝑥 𝑑𝑥 =∫1▒((cos2𝑥 + 1)/2) 𝑑𝑥 = 1/2 ∫1▒(cos2𝑥+1) 𝑑𝑥 = 1/2 [∫1▒cos2𝑥 𝑑𝑥+∫1▒1 𝑑𝑥] We know cos2𝑥=2 cos^2𝑥−1 cos2𝑥+1=2 cos^2𝑥 (cos2𝑥 + 1)/2=cos^2𝑥 ∫1▒𝒄𝒐𝒔𝟐𝒙 𝒅𝒙 Let 2𝑥=𝑡 2 =𝑑𝑡/𝑑𝑥 𝑑𝑥=1/2 𝑑𝑡 ∫1▒cos𝑡 . 1/2 𝑑𝑡 =1/2 (sin𝑡+𝐶1) Putting value of 𝑡 = 2𝑥 =1/2 sin2𝑥+𝐶1 ∫1▒𝟏 𝒅𝒙 =∫1▒𝑥^0 𝑑𝑥 =[𝑥^(0 + 1)/(0 + 1)]+𝐶 =𝑥+𝐶2 Thus, ∫1▒cos^2𝑥 𝑑𝑥=1/2 [∫1▒cos2𝑥 𝑑𝑥+∫1▒1 𝑑𝑥] =1/2 [1/2 sin2𝑥+𝐶1+𝑥+𝐶2] =1/4 sin2𝑥+𝑥/2+1/2(𝐶1+𝐶2) =𝒙/𝟐+𝟏/𝟒 𝐬𝐢𝐧𝟐𝒙+𝑪 ("From (1) and (2) " ) ("Let" 𝐶1+𝐶2=𝐶)
Examples
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Example 1 (iii)
Example 2 (i)
Example 2 (ii)
Example 2 (iii) Important
Example 3 (i)
Example 3 (ii) Important
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Example 4
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Example 5 (iv) Important
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Example 7 (i) You are here
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Example 7 (iii)
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Example 10 (i)
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Example 25 (i)
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About the Author
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