Example 6 - Find integrals (i) sin^3 x cos^2 x dx (ii) sin x

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Example 6 - Chapter 7 Class 12 Integrals - Part 2

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Example 6 - Chapter 7 Class 12 Integrals - Part 3

  1. Chapter 7 Class 12 Integrals (Term 2)
  2. Concept wise

Transcript

Example 6 Find the following integrals: (i) ∫1β–’γ€–sin^3⁑π‘₯ cos^2⁑π‘₯ γ€— 𝑑π‘₯ ∫1β–’γ€–sin^3⁑π‘₯ cos^2⁑π‘₯ γ€— 𝑑π‘₯ Let cos π‘₯=𝑑 Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯. βˆ’sin⁑π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=(βˆ’π‘‘π‘‘)/sin⁑π‘₯ Now are equation becomes ∫1β–’γ€–sin^3⁑π‘₯ cos^2⁑π‘₯ γ€— 𝑑π‘₯ Putting value of π‘π‘œπ‘ β‘π‘₯ and 𝑑π‘₯ = ∫1β–’sin^3⁑π‘₯ .𝑑^2. 𝑑π‘₯ = ∫1β–’sin^3⁑π‘₯ .𝑑^2. 𝑑𝑑/(βˆ’sin⁑π‘₯ ) = ∫1β–’sin^3⁑π‘₯/(βˆ’sin⁑π‘₯ ) 𝑑^2. 𝑑𝑑 = β€“βˆ«1β–’sin^2⁑π‘₯ 𝑑^2. 𝑑𝑑 = – ∫1β–’(1βˆ’cos^2⁑π‘₯ ) 𝑑^2. 𝑑𝑑 = – ∫1β–’(1βˆ’π‘‘^2 ) 𝑑^2. 𝑑𝑑 = – ∫1β–’(𝑑^2βˆ’π‘‘^4 ) 𝑑𝑑 = ∫1β–’(βˆ’π‘‘^2+𝑑^4 ) 𝑑𝑑 = ∫1β–’γ€–βˆ’π‘‘^2 γ€—. 𝑑𝑑 + ∫1▒𝑑^4 . 𝑑𝑑 (∴ sin^2⁑π‘₯=1βˆ’cos^2⁑π‘₯) = (γ€–βˆ’π‘‘γ€—^2+1)/(2 + 1)+𝑑^(4 + 1)/(4 + 1)+𝐢 = (βˆ’π‘‘^3)/3 +𝑑^5/5 +𝐢 Putting back value of t = cos x = (βˆ’πŸ)/πŸ‘ 〖𝒄𝒐𝒔〗^πŸ‘β‘π’™ +𝟏/πŸ“ 〖𝒄𝒐𝒔〗^πŸ“β‘π’™ +π‘ͺ

Chapter 7 Class 12 Integrals (Term 2)
Concept wise

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.