Check sibling questions

Ex 7.4, 15 - Integrate 1 / root (x - a) (x - b) - NCERT Maths

Ex 7.4, 15 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.4, 15 - Chapter 7 Class 12 Integrals - Part 3 Ex 7.4, 15 - Chapter 7 Class 12 Integrals - Part 4 Ex 7.4, 15 - Chapter 7 Class 12 Integrals - Part 5

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


Transcript

Ex 7.4, 15 Integrate the function 1/√((𝑥 − 𝑎)(𝑥 − 𝑏)) ∫1▒1/√((𝑥 − 𝑎) (𝑥 − 𝑏)) 𝑑𝑥 =∫1▒1/√(𝑥(𝑥 − 𝑎) − 𝑎(𝑥 − 𝑏)) 𝑑𝑥 =∫1▒1/√(𝑥^2 − 𝑏𝑥 − 𝑎𝑥 + 𝑎𝑏) 𝑑𝑥 =∫1▒1/√(𝑥^2 − 𝑥(𝑎 + 𝑏) + 𝑎𝑏) 𝑑𝑥 =∫1▒1/√(𝑥^2 − 2(𝑥)((𝑎 + 𝑏)/2) + 𝑎𝑏) 𝑑𝑥 =∫1▒1/√(𝑥^2 − 2(𝑥)((𝑎 + 𝑏)/2) + ((𝑎 + 𝑏)/2)^2− ((𝑎 + 𝑏)/2)^2+ 𝑎𝑏) 𝑑𝑥 =∫1▒1/√((𝑥 − (𝑎 + 𝑏)/2)^2 − ((𝑎 + 𝑏)/2)^2+ 𝑎𝑏) 𝑑𝑥 =∫1▒1/√((𝑥 − (𝑎 + 𝑏)/2)^2 − ((𝑎^2 + 𝑏^2+ 2𝑎𝑏)/4) + 𝑎𝑏) 𝑑𝑥 =∫1▒1/√((𝑥 − (𝑎 + 𝑏)/2)^2+ (− 𝑎^2 − 𝑏^2 − 2𝑎𝑏 + 4𝑎𝑏)/4) 𝑑𝑥 =∫1▒1/√((𝑥 − (𝑎 + 𝑏)/2)^2 + (− 𝑎^2 − 𝑏^2 + 2𝑎𝑏)/4) 𝑑𝑥 =∫1▒1/(√((𝑥 − (𝑎 + 𝑏)/2)^2 − ((𝑎^2 + 𝑏^2 − 2𝑎𝑏)/4) ) ) 𝑑𝑥 =∫1▒1/(√((𝑥 − (𝑎 + 𝑏)/2)^2 − ((𝑎 − 𝑏)/2)^2 ) ) 𝑑𝑥 =𝑙𝑜𝑔⁡|𝑥 − (𝑎 + 𝑏)/2 +√((𝑥 − (𝑎 + 𝑏)/2)^2− ((𝑎 − 𝑏)/2)^2 )|+𝐶 =𝑙𝑜𝑔⁡|𝑥− (𝑎 + 𝑏)/2 +√(𝑥^2+((𝑎 + 𝑏)/2)^2−2(𝑥)((𝑎 + 𝑏)/2)−((𝑎 − 𝑏)/2)^2 )|+𝐶 It is of form ∫1▒𝑑𝑥/√(𝑥^2 − 𝑎^2 ) =𝑙𝑜𝑔⁡|𝑥+√(𝑥^2−𝑎^2 )|+𝐶 ∴ Replacing 𝑥 by (𝑥− (𝑎 + 𝑏)/2) and a by ((𝑎 − 𝑏)/2) , we get =𝑙𝑜𝑔⁡|𝑥− (𝑎 + 𝑏)/2 +√(𝑥^2−2(𝑥)((𝑎 + 𝑏)/2)+((𝑎 + 𝑏)/2)^2−((𝑎 − 𝑏)/2)^2 )|+𝐶 =𝑙𝑜𝑔⁡|𝑥− (𝑎 + 𝑏)/2 +√(𝑥^2−𝑥(𝑎+𝑏)+(𝑎^2 + 𝑏^2 + 2𝑎𝑏)/4−(𝑎^2 + 𝑏^2 − 2𝑎𝑏)/4)|+𝐶 =𝑙𝑜𝑔⁡|𝑥− (𝑎 + 𝑏)/2 +√(𝑥^2−𝑥(𝑎+𝑏)+2𝑎𝑏/4+2𝑎𝑏/4) |+𝐶 =𝑙𝑜𝑔⁡|𝑥− (𝑎 + 𝑏)/2 +√(𝑥^2−𝑥(𝑎+𝑏)+4𝑎𝑏/4) |+𝐶 =𝑙𝑜𝑔⁡|𝑥− (𝑎 + 𝑏)/2 +√(𝑥^2−𝑥(𝑎+𝑏)+𝑎𝑏) |+𝐶 =𝑙𝑜𝑔⁡|𝑥− (𝑎 + 𝑏)/2 +√(𝑥^2−𝑎𝑥−𝑏𝑥+𝑎𝑏) |+𝐶 =𝑙𝑜𝑔⁡|𝑥− (𝑎 + 𝑏)/2 +√(𝑥(𝑥−𝑎)−𝑏(𝑥−𝑎) ) |+𝐶 =𝒍𝒐𝒈⁡|𝒙− (𝒂 + 𝒃)/𝟐 +√((𝒙−𝒂)(𝒙−𝒃) ) |+𝑪

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.