Check sibling questions

Example 16 - Find integral x2 + x + 1 dx / (x + 2) (x2 + 1)

Example 16 - Chapter 7 Class 12 Integrals - Part 2
Example 16 - Chapter 7 Class 12 Integrals - Part 3
Example 16 - Chapter 7 Class 12 Integrals - Part 4
Example 16 - Chapter 7 Class 12 Integrals - Part 5

Solve all your doubts with Teachoo Black (new monthly pack available now!)


Transcript

Example 16 Find ∫1β–’(π‘₯^2+ π‘₯ +1 𝑑π‘₯ )/((π‘₯ + 2) (π‘₯^2+1) ) We can write equation as (π‘₯^2+ π‘₯ + 1)/((π‘₯ + 1) (π‘₯ + 2) )=𝐴/(π‘₯ + 2) + (𝐡π‘₯ + 𝐢)/(π‘₯^2+ 1) Cancelling denominator γ€– π‘₯γ€—^2+ π‘₯+1=𝐴(π‘₯^2+1)+(𝐡π‘₯+𝐢) (π‘₯+2) Putting x = βˆ’πŸ (βˆ’2)^2+(βˆ’2)+1=𝐴((βˆ’2)^2+1)+0 4βˆ’2+1= 5A 3/5 = A Putting x = 𝟎 π‘₯^2+ π‘₯+1=𝐴(π‘₯^2+1)+(𝐡π‘₯+𝐢) (π‘₯+2) 0+0+0= A(0 + 1) + (0 + C) (0 + 2) 1 = A + 2C 1 = 3/5 + 2C 1 – 3/5 = 2C 2/5 = 2C C = 1/5 Putting x = 1 π‘₯^2+ π‘₯+1=𝐴(π‘₯^2+1)+(𝐡π‘₯+𝐢) (π‘₯+2) 1+1+1= 2A + (B + C)(3) 3 = 2A + 3 (B + C) 3 = 2(3/5) + 3 (B+1/5) 3 – 6/5 = 3 (B+1/5) 9/5 = 3 (B+1/5) 3/5 – 1/5 = B B = 2/5 Thus, (π‘₯^2+ π‘₯ + 1)/((π‘₯ + 1) (π‘₯ + 2) )=𝐴/(π‘₯ + 2) + (𝐡π‘₯ + 𝐢)/(π‘₯^2+ 1) (π‘₯^2+ π‘₯ + 1)/((π‘₯ + 1)(π‘₯^2+ 1)) = 3/(5 (π‘₯ + 2)) + (1 (2π‘₯ + 1))/(5 (π‘₯^2 + 1)) Hence, our equation becomes ∫1β–’(π‘₯^2+ π‘₯ + 1)/((π‘₯ + 2) (π‘₯^2 + 1)) 𝑑π‘₯= ∫1β–’3/(5(π‘₯^2 + 1)) 𝑑π‘₯+∫1β–’1/5 ((2π‘₯ + 1))/(π‘₯^2 + 1) 𝑑π‘₯ = ∫1β–’3/(5(π‘₯^2 + 1)) 𝑑π‘₯+ 1/5 ∫1β–’γ€–2π‘₯/(π‘₯^2 + 1) 𝑑π‘₯+γ€— 1/5 ∫1β–’1/(π‘₯^2 + 1) 𝑑π‘₯ 𝐈𝟐 1/5 ∫1β–’2π‘₯/(π‘₯^2+ 1) 𝑑π‘₯ Let 𝑑=π‘₯^2+ 1 𝑑𝑑/𝑑π‘₯=2π‘₯ 𝑑𝑑=2π‘₯ 𝑑π‘₯ Substituting, =1/5 ∫1▒𝑑𝑑/𝑑 = 1/5 log |𝑑| + C_2 = 1/5 log |π‘₯^2+1| + C_2 πˆπŸ‘ 1/5 ∫1β–’1/(π‘₯^2+ 1) 𝑑π‘₯ = 1/5 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (π‘₯)+C_3 Hence ∫1β–’(π‘₯^2+ π‘₯ + 1)/((π‘₯ + 2) (π‘₯^2+ 1)) 𝑑π‘₯ =πŸ‘/πŸ“ π’π’π’ˆ|𝒙+𝟐|+𝟏/πŸ“ π’π’π’ˆ|𝒙^𝟐+𝟏|+𝟏/πŸ“ 〖𝒕𝒂𝒏〗^(βˆ’πŸ) (𝒙)+ C where C = C_1+ C_2+C_3

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.