Example 22 - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
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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Example 1 (ii)
Example 1 (iii)
Example 2 (i)
Example 2 (ii)
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Example 3 (i)
Example 3 (ii) Important
Example 3 (iii)
Example 4
Example 5 (i)
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Example 7 (i)
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Example 8 (i)
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Example 9 (i)
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Example 10 (i)
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About the Author
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo