Chapter 7 Class 12 Integrals
Concept wise

Ex 7.5, 11 - Integrate 5x / (x + 1) (x^2 - 4) - NCERT Maths

Ex 7.5, 11 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.5, 11 - Chapter 7 Class 12 Integrals - Part 3 Ex 7.5, 11 - Chapter 7 Class 12 Integrals - Part 4

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex 7.5, 11 Integrate the function 5π‘₯/((π‘₯ + 1) (π‘₯2βˆ’ 4) ) We can write the integrand as 5π‘₯/((π‘₯ + 1) (π‘₯2βˆ’ 4) ) = 5π‘₯/((π‘₯ + 1) (π‘₯ βˆ’ 2) (π‘₯ + 2) ) 5π‘₯/((π‘₯ + 1) (π‘₯2βˆ’ 4) ) = 𝐴/((π‘₯ + 1) ) + 𝐡/((π‘₯ βˆ’ 2) ) + 𝐢/((π‘₯ + 2) ) 5π‘₯/((π‘₯ + 1) (π‘₯2βˆ’ 4) ) = (𝐴(π‘₯ βˆ’ 2)(π‘₯ + 2) + 𝐡(π‘₯ + 1)(π‘₯ + 2) + 𝐢(π‘₯ +1)(π‘₯ βˆ’ 2))/((π‘₯ + 1) (π‘₯ βˆ’ 2) (π‘₯ + 2) ) Cancelling denominator 5π‘₯ = 𝐴(π‘₯βˆ’2)(π‘₯+2)+𝐡(π‘₯+1)(π‘₯+2)+𝐢(π‘₯+1)(π‘₯βˆ’2) …(1) Putting x = βˆ’1 in (1) 5π‘₯ = 𝐴(π‘₯βˆ’2)(π‘₯+2)+𝐡(π‘₯+1)(π‘₯+2)+𝐢(π‘₯+1)(π‘₯βˆ’2) 5( βˆ’1) = 𝐴(βˆ’1βˆ’2)(βˆ’1+2)+𝐡(βˆ’1+1)(βˆ’1+2)+𝐢(βˆ’1+1)(βˆ’1βˆ’2) βˆ’5 = 𝐴(βˆ’3)(1)+𝐡×0+𝐢×0 βˆ’5 = βˆ’3𝐴 𝐴 = (βˆ’5)/(βˆ’3) = 5/3 Putting x = 2 in (1) 5π‘₯ = 𝐴(π‘₯βˆ’2)(π‘₯+2)+𝐡(π‘₯+1)(π‘₯+2)+𝐢(π‘₯+1)(π‘₯βˆ’2) 5"(2) = " 𝐴(2βˆ’2)(2+2)+𝐡(2+1)(2+2)+𝐢(2+1)(2βˆ’2) 10 = 𝐴×0+𝐡(3)(4)+𝐢×0 10 = 12𝐡 𝐡 = 10/12=5/6 Putting x = βˆ’2 in (1) 5π‘₯ = 𝐴(π‘₯βˆ’2)(π‘₯+2)+𝐡(π‘₯+1)(π‘₯+2)+𝐢(π‘₯+1)(π‘₯βˆ’2) 5"("βˆ’"2) = " 𝐴(βˆ’2βˆ’2)(βˆ’2+2)+𝐡(βˆ’2+1)(βˆ’2+2)+𝐢(βˆ’2+1)(βˆ’2βˆ’2) βˆ’10 = 𝐴×0+𝐡×0+𝐢(βˆ’1)(βˆ’4) βˆ’10 = 4𝐢 𝐢 = (βˆ’10)/4 𝐢 = (βˆ’5)/2 Therefore ∫1β–’5π‘₯/((π‘₯ + 1) (π‘₯2βˆ’ 4) )=∫1β–’(𝐴/(π‘₯ + 1)+𝐡/(π‘₯ βˆ’ 2)+𝐢/(π‘₯ + 2)) 𝑑π‘₯ =5/3 ∫1▒𝑑π‘₯/(π‘₯ + 1) 𝑑π‘₯+ 5/6 ∫1▒𝑑π‘₯/(π‘₯ βˆ’ 2) 𝑑π‘₯βˆ’5/2 ∫1▒𝑑π‘₯/((π‘₯ + 2) ) =πŸ“/πŸ‘ γ€–π’π’π’ˆ 〗⁑|𝒙+𝟏|βˆ’ πŸ“/𝟐 γ€–π₯𝐨𝐠 〗⁑|𝒙+𝟐|+πŸ“/πŸ” γ€–π₯𝐨𝐠 〗⁑|π’™βˆ’πŸ|+π‘ͺ

Ask a doubt
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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.