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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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 It is of form ∫1▒〖𝒅𝒙/√(𝒂^𝟐 βˆ’ 𝒙^𝟐 )=π’”π’Šπ’^(βˆ’πŸ) 𝒙/𝒂+π‘ͺγ€— Replacing π‘₯ by 𝑑 and π‘Ž by 3, we get 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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo