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


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

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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.