Example 25 (ii) - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Examples
Example 1 (i)
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Chapter 7 Class 12 Integrals
Serial order wise
Transcript
Example 25 Evaluate the following integrals: (ii) ∫_4^9▒√𝑥/((30 − 𝑥^(3/2) )^2 ) 𝑑𝑥 Step 1 :- ∫1▒√𝑥/(30 − 𝑥^(3/2) )^2 𝑑𝑥 Let 30−𝑥^(3/2)=𝑡 Differentiating w.r.t. 𝑥 both sides 𝑑(30 − 𝑥^(3/2) )/𝑑𝑥=𝑑𝑡/𝑑𝑥 −3/2 𝑥^(3/2 −1)=𝑑𝑡/𝑑𝑥 (−3)/2 𝑥^(1/2 )=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡/(− 3/2 〖 𝑥〗^( 1/2 ) ) 𝑑𝑥=(−2𝑑𝑡)/(3√𝑥) Therefore, our equation becomes ∫1▒〖(√𝑥 𝑑𝑥 )/(30−𝑥^( 3/2) )^2 =∫1▒〖√𝑥/𝑡^2 (−2 𝑑𝑡)/(3 √𝑥)〗〗 =(−2)/( 3) ∫1▒( 𝑑𝑡)/𝑡^2 =(−2)/( 3) ∫1▒〖𝑡^(−2) 𝑑𝑡〗 =(−2)/( 3) 𝑡^(− 2 + 1)/(− 2 + 1) =(−2)/( 3) 𝑡^(− 1)/(−1) =2/3 𝑡^(−1) =2/3𝑡 Putting 𝑡=(30−𝑥^(3/2) ) =2/3(30 − 𝑥^(3/2) ) Hence F(𝑥)=2/3(30 − 𝑥^(3/2) ) Step 2 :- ∫_4^9▒√𝑥/((30 − 𝑥^( 3/2) ) ) 𝑑𝑥=𝐹(9)−𝐹(4) =2/3(30 − (9)^(3/2) ) −2/3(30 − (4)^(3/2) ) = 2/(3 (30 − (3^2 )^(2/3) ) )−2/(3 (30 − (2^2 )^(2/3) ) ) =2/3 [1/(30 − 3^3 )−1/(30 − 2^3 )] =2/3 [1/(30 − 27)−1/(30 − 8)] =2/3 [1/3−1/22] =2/3 [(22 − 3)/(3 × 22)] =2/3 (19/66) =19/(3 (33)) =𝟏𝟗/𝟗𝟗
