Example 7 (i) - Chapter 7 Class 12 Integrals

Transcript

Example 7 Find (i) ∫1▒cos^2⁡𝑥 𝑑𝑥 ∫1▒cos^2⁡𝑥 𝑑𝑥 =∫1▒((cos⁡2𝑥 + 1)/2) 𝑑𝑥 = 1/2 ∫1▒(cos⁡2𝑥+1) 𝑑𝑥 = 1/2 [∫1▒cos⁡2𝑥 𝑑𝑥+∫1▒1 𝑑𝑥] We know cos⁡2𝑥=2 cos^2⁡𝑥−1 cos⁡2𝑥+1=2 cos^2⁡𝑥 (cos⁡2𝑥 + 1)/2=cos^2⁡𝑥 ∫1▒𝒄𝒐𝒔⁡𝟐𝒙 𝒅𝒙 Let 2𝑥=𝑡 2 =𝑑𝑡/𝑑𝑥 𝑑𝑥=1/2 𝑑𝑡 ∫1▒cos⁡𝑡 . 1/2 𝑑𝑡 =1/2 (sin⁡𝑡+𝐶1) Putting value of 𝑡 = 2𝑥 =1/2 sin⁡2𝑥+𝐶1 ∫1▒𝟏 𝒅𝒙 =∫1▒𝑥^0 𝑑𝑥 =[𝑥^(0 + 1)/(0 + 1)]+𝐶 =𝑥+𝐶2 Thus, ∫1▒cos^2⁡𝑥 𝑑𝑥=1/2 [∫1▒cos⁡2𝑥 𝑑𝑥+∫1▒1 𝑑𝑥] =1/2 [1/2 sin⁡2𝑥+𝐶1+𝑥+𝐶2] =1/4 sin⁡2𝑥+𝑥/2+1/2(𝐶1+𝐶2) =𝒙/𝟐+𝟏/𝟒 𝐬𝐢𝐧⁡𝟐𝒙+𝑪 ("From (1) and (2) " ) ("Let" 𝐶1+𝐶2=𝐶)