Ex 7.3, 9 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.3, 9 Integrate (𝑐𝑜𝑠 𝑥)/(1 + 𝑐𝑜𝑠 𝑥) ∫1▒〖(𝑐𝑜𝑠 𝑥)/(1 + 𝑐𝑜𝑠 𝑥) " " 𝑑𝑥〗 = ∫1▒((cos⁡𝑥 + 1 − 1)/(1 + cos⁡𝑥 )) 𝑑𝑥 =∫1▒((1 + cos⁡𝑥 − 1)/(1 + cos⁡𝑥 )) 𝑑𝑥 =∫1▒((1 + cos⁡𝑥)/(1 + cos⁡𝑥 ) − 1/(1 + cos⁡𝑥 )) 𝑑𝑥 =∫1▒〖1−1/(1 + cos⁡𝑥 )〗 𝑑𝑥 =∫1▒1 𝑑𝑥−∫1▒𝟏/(𝟏 + 𝒄𝒐𝒔⁡𝒙 ) 𝑑𝑥 =∫1▒1 𝑑𝑥−∫1▒1/(𝟐 〖𝒄𝒐𝒔〗^𝟐⁡〖𝒙/𝟐〗 ) 𝑑𝑥 =∫1▒1 𝑑𝑥−∫1▒1/2 sec^2⁡〖𝑥/2〗 𝑑𝑥 =∫1▒1 𝑑𝑥−1/2 ∫1▒sec^2⁡〖𝑥/2〗 𝑑𝑥 =𝑥− 1/2 〖tan 〗⁡〖𝑥/2〗/(1/2) +𝐶 =𝑥− 2/2 〖tan 〗⁡〖𝑥/2〗 +𝐶 =𝒙− 〖𝐭𝐚𝐧 〗⁡〖𝒙/𝟐〗 +𝑪 ∫1▒sec^2⁡(𝑎𝑥+𝑏) 𝑑𝑥=𝑡𝑎𝑛⁡(𝑎𝑥 + 𝑏)/𝑎 +𝐶 We know that cos 2𝜃=2 cos^2⁡〖𝜃−1〗 cos⁡2𝜃+1=2 cos^2⁡𝜃 Replacing 𝜃 by 𝑥/2 cos⁡2(𝑥/2)+1=2 cos^2⁡〖𝑥/2〗 cos⁡𝑥+1=2 cos^2⁡〖𝑥/2〗