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Example 13 - Find integral 3x - 2 / (x + 1)2 (x + 3) dx - Examples

Example 13 - Chapter 7 Class 12 Integrals - Part 2
Example 13 - Chapter 7 Class 12 Integrals - Part 3
Example 13 - Chapter 7 Class 12 Integrals - Part 4
Example 13 - Chapter 7 Class 12 Integrals - Part 5

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Transcript

Example 13 Find ∫1β–’(3π‘₯ βˆ’2)/((π‘₯ + 1)^2 (π‘₯ + 3) ) 𝑑π‘₯ We can write Integral as (3π‘₯ βˆ’ 2)/((π‘₯ + 1)^2 (π‘₯ + 3) )=𝐴/(π‘₯ + 1) + 𝐡/(π‘₯ + 1)^2 + 𝐢/((π‘₯ + 3) ) (3π‘₯ βˆ’ 2)/((π‘₯ + 1)^2 (π‘₯ + 3) )=(𝐴(π‘₯ + 1)(π‘₯ + 3) + 𝐡(π‘₯ + 3) + 𝐢(π‘₯ + 1)^2)/((π‘₯ + 1)^2 (π‘₯ + 3) ) Cancelling denominator 3π‘₯ βˆ’2=𝐴(π‘₯+1)(π‘₯+3)+𝐡(π‘₯+3)+𝐢(π‘₯+1)^2 Putting x = βˆ’1 3(βˆ’1) βˆ’2=𝐴(βˆ’1+1)(βˆ’1+3)+𝐡(βˆ’1+3)+𝐢(βˆ’1+1)^2 βˆ’3βˆ’2=𝐴×0+𝐡×2+𝐢×(0)^2 βˆ’5=𝐡×2 𝐡=(βˆ’ 5)/2 Putting x = βˆ’ 3 3π‘₯ βˆ’2=𝐴(π‘₯+1)(π‘₯+3)+𝐡(π‘₯+3)+𝐢(π‘₯+1)^2 3(βˆ’3)βˆ’2=𝐴(βˆ’3+1)(βˆ’3+3)+𝐡(βˆ’3+3)+𝐢(βˆ’3+1)^2 βˆ’9βˆ’2=𝐴×0+𝐡×0+𝐢×(βˆ’2)^2 βˆ’11=0+0+𝐢(4) βˆ’11=4𝐢 (βˆ’11)/4 =𝐢 𝐢 =(βˆ’11)/4 Putting x = 0 3π‘₯ βˆ’2=𝐴(π‘₯+1)(π‘₯+3)+𝐡(π‘₯+3)+𝐢(π‘₯+1)^2 3(0) βˆ’ 2 = A(1) (3) + B(3) + C γ€–"(1)" γ€—^2 βˆ’2 = 3A + 3B + C Putting value of B & C βˆ’2 = 3A + 3((βˆ’5)/2) + ((βˆ’11)/4) βˆ’2 = 3A βˆ’ 15/2βˆ’11/4 βˆ’2 = 3A + (βˆ’30 βˆ’ 11)/4 βˆ’8 = 12A βˆ’ 41 41 βˆ’ 8 = 12A 33/12 = A 11/4 = A Hence, we can write (3π‘₯ βˆ’ 2)/((π‘₯ + 1)^2 (π‘₯ + 3) )=𝐴/(π‘₯ + 1) + 𝐡/(π‘₯ + 1)^2 + 𝐢/((π‘₯ + 3) ) (3π‘₯ βˆ’ 2)/((π‘₯ + 1)^2 (π‘₯ + 3) )=11/4(π‘₯ + 1) βˆ’ 5/γ€–2(π‘₯ + 1)γ€—^2 βˆ’ 11/4(π‘₯ + 3) Therefore ∫1β–’(3π‘₯ βˆ’ 2)/((π‘₯ + 1)^2 (π‘₯ + 3) ) 𝑑π‘₯ =∫1β–’11/4(π‘₯ + 1) 𝑑π‘₯βˆ’βˆ«1β–’5/γ€–2(π‘₯ + 1)γ€—^2 𝑑π‘₯βˆ’βˆ«1β–’11/4(π‘₯ + 3) 𝑑π‘₯ =11/4 log⁑|π‘₯+1|βˆ’5/2Γ—((βˆ’1))/((π‘₯ + 1) ) βˆ’ 11/4 log⁑|π‘₯+3|+𝐢 =11/4 (log⁑|π‘₯+1|βˆ’log⁑|π‘₯+3| )+5/(2 (π‘₯ + 1) )+𝐢 =𝟏𝟏/πŸ’ π’π’π’ˆβ‘|(𝒙 + 𝟏)/(𝒙 + πŸ‘)| + πŸ“/(𝟐 (𝒙 + 𝟏) )+π‘ͺ

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.