Ex 7.6, 18 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.6, 18 Integrate the function - 𝑒𝑥 ((1 + sin⁡𝑥)/(1 + cos⁡𝑥 )) Simplifying function 𝑒^𝑥 ((1 + sin⁡𝑥)/(1 + cos⁡𝑥 )) 𝑒^𝑥 ((1 + sin⁡𝑥)/(1 + cos⁡𝑥 ))=𝑒^𝑥 ((1 + 2 sin⁡(𝑥/2) cos⁡(𝑥/2))/(2 〖𝑐𝑜𝑠^2〗⁡(𝑥/2) )) 𝒔𝒊𝒏⁡𝟐𝒙=𝟐 𝒔𝒊𝒏⁡𝒙 𝒄𝒐𝒔⁡𝒙 Replacing x by 𝑥/2 , we get 〖𝑠𝑖𝑛 2〗⁡(𝑥/2)=2 𝑠𝑖𝑛⁡(𝑥/2) 𝑐𝑜𝑠⁡(𝑥/2) 𝑠𝑖𝑛⁡𝑥=2 𝑠𝑖𝑛⁡(𝑥/2) 𝑐𝑜𝑠⁡(𝑥/2) 𝐜𝐨𝐬⁡𝟐𝒙= 2〖"cos" 〗^𝟐 𝜽−𝟏 Replacing 𝜃𝑥 by 𝑥/2 〖𝑐𝑜𝑠 2〗⁡(𝑥/2)=2𝑐𝑜𝑠^2 (𝑥/2)−1 1+𝑐𝑜𝑠⁡𝑥=2𝑐𝑜𝑠^2 𝑥/2 We know 〖𝑠𝑖𝑛^2〗⁡𝑥+〖𝑐𝑜𝑠^2〗⁡𝑥⁡= 1 Replacing x by 𝑥/2 , we get 〖𝑠𝑖𝑛^2〗⁡(𝑥/2)⁡〖+〖𝑐𝑜𝑠^2〗⁡(𝑥/2) 〗=1 =𝑒^𝑥 ((〖𝒔𝒊𝒏〗^𝟐⁡(𝒙/𝟐) + 〖𝒄𝒐𝒔〗^𝟐⁡(𝒙/𝟐) +2 〖sin 〗⁡〖𝑥/2〗 . 〖cos 〗⁡〖𝑥/2〗)/( 2 〖𝑐𝑜𝑠^2〗⁡(𝑥/2) )) =𝑒^𝑥 ((〖sin 〗⁡〖𝑥/2〗 + 〖cos 〗⁡〖𝑥/2〗 )^2/( 2 〖𝑐𝑜𝑠^2〗⁡(𝑥/2) )) =𝑒^𝑥/2 ((〖sin 〗⁡〖𝑥/2〗 + 〖cos 〗⁡〖𝑥/2〗)/( 𝑐𝑜𝑠⁡〖 𝑥/2〗 ))^2 =𝑒^𝑥/2 (〖sin 〗⁡〖𝑥/2〗/( 𝑐𝑜𝑠⁡〖 𝑥/2〗 ) + 〖cos 〗⁡〖𝑥/2〗/( 𝑐𝑜𝑠⁡〖 𝑥/2〗 ))^2 =𝑒^𝑥/2 (〖tan 〗⁡〖𝑥/2〗 +1)^2 =𝑒^𝑥/2 (tan^2⁡〖𝑥/2〗 +(1)^2+2(〖tan 〗⁡〖𝑥/2〗 )(1)) =𝑒^𝑥/2 (〖〖𝒕𝒂𝒏〗^𝟐 〗⁡〖𝒙/𝟐〗 +𝟏+2 〖tan 〗⁡〖𝑥/2〗 ) =𝑒^𝑥/2 (〖〖𝒔𝒆𝒄〗^𝟐 〗⁡〖𝒙/𝟐〗 +2 〖tan 〗⁡(𝑥/2) ) =𝑒^𝑥 (1/2 . sec^2⁡〖𝑥/2〗 +2/2 〖tan 〗⁡(𝑥/2) ) =𝑒^𝑥 (〖tan 〗⁡(𝑥/2)+1/2 sec^2⁡(𝑥/2) ) We know 〖1+tan^2〗⁡𝑥=sec^2⁡𝑥 Replacing x by 𝑥/2 , we get 〖1+〖𝑡𝑎𝑛^2〗⁡(𝑥/2)〗⁡= sec^2 (𝑥/2) Thus, Our Integration becomes ∫1▒〖𝑒^𝑥 " " ((1 + sin⁡𝑥)/(1 + cos⁡𝑥 )) 𝑑𝑥〗 " "=∫1▒〖𝑒^𝑥 (〖tan 〗⁡(𝑥/2)+1/2 sec^2⁡(𝑥/2) )𝑑𝑥〗 " " It is of the form ∫1▒〖𝑒^𝑥 [𝑓(𝑥)+𝑓^′ (𝑥)] 〗 𝑑𝑥=𝑒^𝑥 𝑓(𝑥)+𝐶 Where 𝑓(𝑥)=tan^(−1)⁡𝑥 𝑓^′ (𝑥)= 1/(1 + 𝑥^2 ) So, our equation becomes ∫1▒〖𝑒^𝑥 " " ((1 + sin⁡𝑥)/(1 + cos⁡𝑥 )) 𝑑𝑥〗 " " = 𝒆^𝒙 〖𝒕𝒂𝒏 〗⁡〖𝒙/𝟐〗+𝑪