Ex 7.2, 7 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.2, 7 Integrate the function: 𝑥√(𝑥+2) 𝑥√(𝑥+2) Step 1: Let (𝑥+2)=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 1+0 = 𝑑𝑡/𝑑𝑥 1= 𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡 Step 2: Integrating the function ∫1▒〖" " 𝑥√(𝑥+2)〗 .𝑑𝑥 Putting the value of 𝑥+2 & 𝑑𝑥 . = ∫1▒〖𝑥√𝑡〗 .𝑑𝑥 = ∫1▒〖𝑥√𝑡〗 .𝑑𝑡 = ∫1▒〖(𝑡−2) √𝑡〗 .𝑑𝑡 = ∫1▒〖(𝑡−2) 𝑡^(1/2) 〗 .𝑑𝑡 = ∫1▒(𝑡.𝑡^(1/2)−2.𝑡^(1/2) ) .𝑑𝑡 = ∫1▒(𝑡^(3/2)−2.𝑡^(1/2) ) .𝑑𝑡 = ∫1▒𝑡^(3/2) .𝑑𝑡 − 2∫1▒𝑡^(1/2) .𝑑𝑡 (Using 𝑥+2=𝑡, 𝑥=𝑡−2) = 𝑡^(3/2 + 1)/(3/2 + 1) − 2 . 𝑡^(1/2 + 1)/(1/2 + 1) + 𝐶 = 𝑡^(5/2)/(5/2) − 2 . 𝑡^(3/2)/(3/2) + 𝐶 = 2/5 𝑡^(5/2) − 2 × 2/3 𝑡^(3/2) + 𝐶 = 2/5 𝑡^(5/2) − 4/3 𝑡^(3/2) + 𝐶 Putting back 𝑡=𝑥+2 = 𝟐/𝟓 (𝒙+𝟐)^(𝟓/𝟐) − 𝟒/𝟑 (𝒙+𝟐)^(𝟑/𝟐) + 𝑪 (Using ∫1▒𝑥^𝑛 . 𝑑𝑥=𝑥^(𝑛+1)/(𝑛 +1) )