Find f sin 2 x/ 9 - cos4 x dx

This question is similar to Example 42 - Chapter 7 Class 12 - Integrals

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  1. Class 12
  2. Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Transcript

Question 1 – Choice 2 Find ∫1β–’γ€–sin⁑2π‘₯/√(9 βˆ’ cos^4⁑π‘₯ ) 𝑑π‘₯γ€— Now, ∫1β–’γ€–sin⁑2π‘₯/√(9 βˆ’ cos^4⁑π‘₯ ) 𝑑π‘₯γ€—=∫1β–’γ€–(𝟐 𝐬𝐒𝐧⁑𝒙 πœπ¨π¬β‘π’™)/√(πŸ— βˆ’(〖𝒄𝒐𝒔〗^πŸβ‘π’™ )^𝟐 ) 𝒅𝒙〗 Let 〖𝒄𝒐𝒔〗^πŸβ‘π’™=𝒕 Differentiating both sides w.r.t.π‘₯ 2 cos⁑π‘₯ Γ— βˆ’sin⁑π‘₯=𝑑𝑑/𝑑π‘₯ 𝒅𝒙=𝒅𝒕/(β€“πŸ 𝒄𝒐𝒔⁑𝒙 π’”π’Šπ’β‘π’™ ) Hence, our equation becomes ∫1β–’γ€–(𝟐 𝐬𝐒𝐧⁑𝒙 πœπ¨π¬β‘π’™)/√(πŸ— βˆ’(〖𝒄𝒐𝒔〗^πŸβ‘π’™ )^𝟐 ) 𝒅𝒙〗 =∫1β–’γ€–(2 sin⁑π‘₯ cos⁑π‘₯)/√(9 βˆ’ 𝑑^2 ) 𝑑π‘₯γ€— =∫1β–’γ€–(2 sin⁑π‘₯ cos⁑π‘₯)/√(9 βˆ’ 𝑑^2 )×𝒅𝒕/(β€“πŸ 𝒄𝒐𝒔⁑𝒙 π’”π’Šπ’β‘π’™ )γ€— =βˆ’βˆ«1▒𝑑𝑑/√((3)^2 βˆ’ (𝑑)^2 ) =βˆ’[sin^(βˆ’1)⁑〖𝑑/3+𝐢1γ€— ] =βˆ’π’”π’Šπ’^(βˆ’πŸ) 𝒕/πŸ‘+π‘ͺ Putting back 𝒕=〖𝒄𝒐𝒔〗^πŸβ‘π’™ =βˆ’π’”π’Šπ’^(βˆ’πŸ) [𝟏/πŸ‘ 𝒄𝒐𝒔^𝟐 𝒙]+π‘ͺ

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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.