Example 22 - Find (i) ex ( tan-1 x + 1 / 1 + x2) dx - Examples

Example 22 - Chapter 7 Class 12 Integrals - Part 2
Example 22 - Chapter 7 Class 12 Integrals - Part 3 Example 22 - Chapter 7 Class 12 Integrals - Part 4 Example 22 - Chapter 7 Class 12 Integrals - Part 5

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Example 22 Find (i) ∫1▒𝑒^𝑥 (tan^(−1)⁡𝑥+ 1/(1 + 𝑥^2 )) 𝑑𝑥 ∫1▒〖𝑒^𝑥 (tan^(−1)⁡𝑥+1/(1 + 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 (tan^(−1)⁡𝑥+1/(1 + 𝑥^2 ))𝑑𝑥〗=𝒆^𝒙 〖𝐭𝐚𝐧〗^(−𝟏)⁡〖𝒙+𝑪〗 Example 22 Find (ii) ∫1▒((𝑥^2 + 1) 𝑒^𝑥)/(𝑥 + 1)^2 𝑑𝑥 ∫1▒〖(𝑥^2 + 1)/(𝑥 + 1)^2 .𝑒^𝑥 𝑑𝑥〗 Adding and subtracting 1 in numerator =∫1▒〖(𝑥^2+ 1 + 1 − 1)/(𝑥 + 1)^2 .𝑒^𝑥 .𝑑𝑥〗 =∫1▒〖(𝑥^2 − 1 + 1 + 1)/(𝑥 + 1)^2 .𝑒^𝑥 .𝑑𝑥〗 =∫1▒〖[(𝑥^2 − 1)/(𝑥 + 1)^2 +2/(𝑥 + 1)^2 ] 𝑒^𝑥 𝑑𝑥〗 =∫1▒〖𝑒^𝑥 [(𝑥^2 − (1)^2)/(𝑥 + 1)^2 +2/(𝑥 + 1)^2 ]𝑑𝑥〗 =∫1▒〖𝑒^𝑥 [(𝑥 − 1)(𝑥 + 1)/(𝑥 + 1)^2 +2/(𝑥 + 1)^2 ]𝑑𝑥〗 =∫1▒〖𝑒^𝑥 [(𝑥 − 1)/(𝑥 + 1)+2/(𝑥 + 1)^2 ]𝑑𝑥〗 It is of form ∫1▒〖𝑒^𝑥 [𝑓(𝑥)+𝑓^′ (𝑥)] 〗 𝑑𝑥=𝑒^𝑥 𝑓(𝑥)+𝐶 Where 𝑓(𝑥)=(𝑥 − 1)/(𝑥 + 1) 𝑓^′ (𝑥)=𝑑/𝑑𝑥 [(𝑥 − 1)/(𝑥 + 1)] 𝑓^′ (𝑥)=(1.(𝑥 + 1) −1 (𝑥 − 1))/(𝑥 + 1)^2 =(𝑥 + 1 − 𝑥 + 1)/(𝑥 + 1)^2 =2/(𝑥 + 1)^2 Thus, our equation becomes ∫1▒〖(𝑥^2 + 1)/(𝑥 + 1)^2 .𝑒^𝑥=∫1▒〖𝑒^𝑥 [(𝑥 − 1)/(𝑥 + 1)+2/(𝑥 + 1)^2 ]𝑑𝑥〗〗 =𝑒^𝑥 [(𝑥 − 1)/(𝑥 + 1)]+𝐶 =(𝒙 − 𝟏)/(𝒙 + 𝟏).𝒆^𝒙+𝑪

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.