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  1. Class 12
  2. Important Question for exams Class 12
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Misc 18 Integrate the function 1/√(sin^3⁑π‘₯ sin⁑(π‘₯ + 𝛼) ) Consider sin^3⁑π‘₯ sin⁑(π‘₯ + 𝛼) =sin^3⁑π‘₯ [sin⁑π‘₯ cos⁑𝛼+cos⁑π‘₯.sin⁑𝛼 ] =sin^4⁑cos⁑𝛼 +cos⁑π‘₯.sin^3⁑π‘₯ sin⁑𝛼 =sin^4⁑π‘₯ [cos⁑𝛼+cos⁑π‘₯/sin⁑π‘₯ . sin⁑π‘₯ ] =sin^4⁑π‘₯ [cos⁑𝛼+cot⁑π‘₯ . sin⁑π‘₯ ] Hence sin^3⁑π‘₯ sin⁑(π‘₯+𝛼)=sin^4⁑π‘₯ (cos⁑𝛼+cot⁑π‘₯.sin⁑𝛼 ) Now ∫1β–’1/√(sin^3⁑π‘₯ sin⁑(π‘₯ + 𝛼) ) 𝑑π‘₯ =∫1β–’1/√(sin^4⁑π‘₯ (cos⁑𝛼 + cot⁑π‘₯ . sin⁑𝛼 ) ) 𝑑π‘₯ =∫1β–’γ€–1/√(sin^4⁑π‘₯ )Γ—1/√(cos⁑𝛼 + cot⁑π‘₯ . sin⁑𝛼 )γ€— 𝑑π‘₯ =∫1β–’γ€–1/sin^2⁑π‘₯ Γ—1/√(cos⁑𝛼 + cot⁑π‘₯ . sin⁑𝛼 )γ€— 𝑑π‘₯ Let cos⁑𝛼+cot⁑π‘₯. sin⁑𝛼=𝑑 Diff w.r.t. x 𝑑(cos⁑𝛼 + cot⁑π‘₯ sin⁑𝛼 )/𝑑π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑(cos⁑𝛼 )/𝑑π‘₯+sin⁑𝛼 𝑑(cot⁑π‘₯ )/𝑑π‘₯=𝑑𝑑/𝑑π‘₯ 0+sin⁑𝛼 (βˆ’π‘π‘œπ‘ π‘’π‘^2 π‘₯)=𝑑𝑑/𝑑π‘₯ βˆ’sin⁑𝛼 π‘π‘œπ‘ π‘’π‘^2 π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑/(βˆ’sin⁑𝛼 π‘π‘œπ‘ π‘’π‘^2 π‘₯) 𝑑π‘₯=1/(βˆ’sin⁑𝛼 ) . 1/(π‘π‘œπ‘ π‘’π‘^2 π‘₯) . 𝑑𝑑 𝑑π‘₯=1/(βˆ’sin⁑𝛼 ) . sin^2⁑π‘₯. 𝑑𝑑 Now our equation becomes ∫1β–’1/(sin^2⁑π‘₯ √(cos⁑𝛼 + cot⁑π‘₯ sin⁑𝛼 ) ) 𝑑π‘₯=∫1β–’1/(sin^2⁑π‘₯ βˆšπ‘‘ )Γ—1/(βˆ’sin⁑𝛼 )Γ—sin^2⁑π‘₯ 𝑑𝑑 =1/(βˆ’sin⁑𝛼 ) ∫1β–’1/βˆšπ‘‘ 𝑑𝑑 =(βˆ’1)/sin⁑𝛼 ∫1β–’(𝑑)^((βˆ’1)/2) 𝑑𝑑 =(βˆ’1)/sin⁑𝛼 [𝑑^((βˆ’1)/2 + 1)/((βˆ’1)/2 + 1) +𝐢] =(βˆ’1)/sin⁑𝛼 [𝑑^(1/2)/(1/2) +𝐢] =(βˆ’πŸ)/π’”π’Šπ’β‘πœΆ [πŸβˆšπ’• +π‘ͺ] Putting value of 𝑑=√(π‘π‘œπ‘ β‘π›Ό+π‘π‘œπ‘‘β‘π‘₯. 𝑠𝑖𝑛⁑𝛼 ) =(βˆ’1)/sin⁑𝛼 [2√(π‘π‘œπ‘ β‘π›Ό+π‘π‘œπ‘‘β‘π‘₯. 𝑠𝑖𝑛⁑𝛼 )+𝐢] =(βˆ’2)/sin⁑𝛼 √(π‘π‘œπ‘ β‘π›Ό+π‘π‘œπ‘‘β‘π‘₯. 𝑠𝑖𝑛⁑𝛼 ) βˆ’1/sin⁑𝛼 . 𝐢 =(βˆ’2)/sin⁑𝛼 √(π‘π‘œπ‘ β‘π›Ό+π‘π‘œπ‘‘β‘π‘₯. 𝑠𝑖𝑛⁑𝛼 ) +𝐢 From (1) sin^3⁑π‘₯ sin⁑(π‘₯+𝛼)=sin^4⁑π‘₯ (π‘π‘œπ‘ β‘π›Ό+π‘π‘œπ‘‘β‘π‘₯. 𝑠𝑖𝑛⁑𝛼 ) (sin^3⁑π‘₯ sin⁑(π‘₯ + 𝛼))/sin^4⁑π‘₯ =π‘π‘œπ‘ β‘π›Ό+π‘π‘œπ‘‘β‘π‘₯. 𝑠𝑖𝑛⁑𝛼 〖𝑠𝑖𝑛 〗⁑(π‘₯ + 𝛼)/sin⁑π‘₯ =π‘π‘œπ‘ β‘π›Ό+π‘π‘œπ‘‘β‘π‘₯. 𝑠𝑖𝑛⁑𝛼 Thus, Answer =(βˆ’πŸ)/π’”π’Šπ’β‘π’™ √(π’”π’Šπ’β‘(𝒙 + 𝜢)/𝐬𝐒𝐧⁑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 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
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