Ex 7.2, 20 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.2, 20 Integrate the function (𝑒^2𝑥 − 𝑒^(−2𝑥))/(𝑒^2𝑥 + 𝑒^(−2𝑥) ) Let 𝑒^2𝑥 + 𝑒^(−2𝑥)= 𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑒^2𝑥. 𝑑(2𝑥)/𝑑𝑥 +𝑒^(−2𝑥) 𝑑(−2𝑥)/𝑑𝑥= 𝑑𝑡/𝑑𝑥 〖2𝑒〗^2𝑥−〖2𝑒〗^(−2𝑥)= 𝑑𝑡/𝑑𝑥 2(𝑒^2𝑥−𝑒^(−2𝑥) )=𝑑𝑡/𝑑𝑥 " " 𝑑𝑥 = 𝑑𝑡/2(𝑒^2𝑥− 𝑒^(−2𝑥) ) Integrating the function ∫1▒〖" " (𝑒^2𝑥 − 𝑒^(−2𝑥))/(𝑒^2𝑥 + 𝑒^(−2𝑥) )〗. 𝑑𝑥 Putting 𝑒^2𝑥 + 𝑒^(−2𝑥)=𝑡 & 𝑑𝑥=𝑑𝑡/2(𝑒^2𝑥− 𝑒^(−2𝑥) ) = ∫1▒〖" " (𝑒^2𝑥 − 𝑒^(−2𝑥))/𝑡〗. 𝑑𝑡/2(𝑒^2𝑥− 𝑒^(−2𝑥) ) = ∫1▒〖" " 1/2𝑡〗. 𝑑𝑡 = 1/2 ∫1▒1/𝑡. 𝑑𝑡 = 1/2 log⁡〖 |𝑡|〗+𝐶 = 1/2 log⁡〖 |𝑒^2𝑥 + 𝑒^(−2𝑥) |〗+𝐶 = 𝟏/𝟐 𝒍𝒐𝒈⁡〖 (𝒆^𝟐𝒙 + 𝒆^(−𝟐𝒙) )〗+𝑪 (Using ∫1▒1/𝑥. 𝑑𝑥=𝑙𝑜𝑔⁡|𝑥| ) (Using 𝑡=𝑒^2𝑥 + 𝑒^(−2𝑥)) (∴ 𝑒^2𝑥+𝑒^(−2𝑥)>0 )