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Example 27 - Chapter 7 Class 12 Integrals - Part 2

Example 27 - Chapter 7 Class 12 Integrals - Part 3
Example 27 - Chapter 7 Class 12 Integrals - Part 4 Example 27 - Chapter 7 Class 12 Integrals - Part 5 Example 27 - Chapter 7 Class 12 Integrals - Part 6

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Transcript

Example 27 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

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