Ex 7.4
Ex 7.4, 2 Important
Ex 7.4, 3
Ex 7.4, 4
Ex 7.4, 5 Important
Ex 7.4, 6
Ex 7.4, 7
Ex 7.4, 8 Important
Ex 7.4, 9
Ex 7.4, 10
Ex 7.4, 11 Important
Ex 7.4, 12
Ex 7.4, 13 Important
Ex 7.4, 14
Ex 7.4, 15 Important
Ex 7.4, 16
Ex 7.4, 17 Important
Ex 7.4, 18
Ex 7.4, 19 Important
Ex 7.4, 20
Ex 7.4, 21 Important
Ex 7.4, 22
Ex 7.4, 23 Important
Ex 7.4, 24 (MCQ)
Ex 7.4, 25 (MCQ) Important
Last updated at April 16, 2024 by Teachoo
Ex 7.4, 1 (3𝑥^2)/(𝑥^6 + 1) We need to find ∫1▒(𝟑𝒙^𝟐)/(𝒙^𝟔 + 𝟏) 𝒅𝒙 Let 𝒙^𝟑=𝒕 Diff both sides w.r.t. x 3𝑥^2=𝑑𝑡/𝑑𝑥 𝒅𝒙=𝒅𝒕/(𝟑𝒙^𝟐 ) Thus, our equation becomes ∫1▒(𝟑𝒙^𝟐)/(𝒙^𝟔 + 𝟏) 𝒅𝒙 =∫1▒(3𝑥^2)/((𝑥^3 )^2 + 1) 𝑑𝑥 Putting the value of 𝑥^3=𝑡 and 𝑑𝑥=𝑑𝑡/(3𝑥^2 ) =∫1▒(3𝑥^2)/(𝑡^2 + 1) .𝑑𝑡/(3𝑥^2 ) =∫1▒𝑑𝑡/(𝑡^2 + 1) =∫1▒𝒅𝒕/(𝒕^𝟐 + (𝟏)^𝟐 ) =1/1 tan^(−1)〖 𝑡/1 〗+𝐶 It is of form ∫1▒𝑑𝑡/(𝑥^2 + 𝑎^2 ) =1/𝑎 〖〖𝑡𝑎𝑛〗^(−1) 〗〖𝑥/𝑎〗 +𝐶 ∴ Replacing 𝑥 = 𝑡 and 𝑎 by 1 , we get =tan^(−1)〖 (𝑡)〗+𝐶 =〖〖𝒕𝒂𝒏〗^(−𝟏) 〗(𝒙^𝟑 )+𝑪 ("Using" 𝑡=𝑥^3 )