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Ex 7.2, 32  - Integrate 1 / (1 + cot x) - Chapter 7 Class 12

Ex 7.2, 32 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.2, 32 - Chapter 7 Class 12 Integrals - Part 3 Ex 7.2, 32 - Chapter 7 Class 12 Integrals - Part 4

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Ex 7.2, 32 Integrate 1/(1 + cot⁑π‘₯ ) Simplify the given function ∫1β–’1/(1 + cot⁑π‘₯ ) 𝑑π‘₯ = ∫1β–’1/(1 + cos⁑π‘₯/sin⁑π‘₯ ) 𝑑π‘₯ = ∫1β–’1/(γ€–sin⁑π‘₯ + cos〗⁑π‘₯/sin⁑π‘₯ ) 𝑑π‘₯ = ∫1β–’sin⁑π‘₯/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ 𝑑π‘₯ Multiplying & dividing by 2 = ∫1β–’(2 sin⁑π‘₯)/2(γ€–sin⁑π‘₯ + cos〗⁑π‘₯ ) 𝑑π‘₯ Adding & subtracting π‘π‘œπ‘ β‘π‘₯ in numerator = ∫1β–’(sin⁑π‘₯ + sin⁑π‘₯ + cos⁑π‘₯ βˆ’ cos⁑π‘₯)/2(γ€–sin⁑π‘₯ + cos〗⁑π‘₯ ) 𝑑π‘₯ = 1/2 ∫1β–’((sin⁑π‘₯ + cos⁑π‘₯ + sin⁑π‘₯ βˆ’ cos⁑π‘₯)/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ ) 𝑑π‘₯ = 1/2 ∫1β–’((sin⁑π‘₯ + cos⁑π‘₯)/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ +(sin⁑π‘₯ βˆ’ cos⁑π‘₯)/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ ) 𝑑π‘₯ = 1/2 ∫1β–’(1+(sin⁑π‘₯ βˆ’ cos⁑π‘₯)/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ ) 𝑑π‘₯ = 1/2 [π‘₯+∫1β–’((sin⁑π‘₯ βˆ’ cos⁑π‘₯)/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ ) 𝑑π‘₯] + 𝐢1 …(1) Solving 𝐈1 I1 = ∫1β–’(sin⁑π‘₯ βˆ’ cos⁑π‘₯)/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ 𝑑π‘₯ Let γ€–sin⁑π‘₯ + cos〗⁑π‘₯=𝑑 Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ γ€–cos⁑π‘₯βˆ’sin〗⁑π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑/γ€–cos⁑π‘₯ βˆ’ sin〗⁑π‘₯ 𝑑π‘₯=𝑑𝑑/(βˆ’(γ€–sin⁑π‘₯ βˆ’ cos〗⁑π‘₯ ) ) Thus, our equation becomes …(2) I1 = ∫1β–’(sin⁑π‘₯ βˆ’ cos⁑π‘₯)/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ 𝑑π‘₯ = ∫1β–’(sin⁑π‘₯ βˆ’ cos⁑π‘₯)/𝑑 . 𝑑𝑑/(βˆ’(γ€–sin⁑π‘₯ βˆ’ cos〗⁑π‘₯ ) ) = βˆ’1∫1▒𝑑𝑑/𝑑 = βˆ’γ€–log 〗⁑|𝑑|+𝐢 Putting back 𝑑=𝑠𝑖𝑛⁑π‘₯+π‘π‘œπ‘ β‘π‘₯ = βˆ’log⁑〖 |sin⁑π‘₯+cos⁑π‘₯ |γ€—+𝐢2 Putting the value of I1 in (1) ∴ ∫1β–’γ€–1/(1 + cot⁑π‘₯ ) " " γ€— = 1/2 [π‘₯+∫1β–’((sin⁑π‘₯ βˆ’ cos⁑π‘₯)/γ€–sin⁑π‘₯ + cos〗⁑π‘₯ ) 𝑑π‘₯] + 𝐢1 = 1/2 [π‘₯βˆ’log⁑|sin⁑π‘₯+cos⁑π‘₯ |+𝐢2" " ] +𝐢1 = π‘₯/2βˆ’1/2 log⁑〖 |sin⁑π‘₯+cos⁑π‘₯ |γ€—+𝐢1+𝐢2/2 = 𝒙/𝟐 βˆ’πŸ/𝟐 π’π’π’ˆβ‘γ€– |π’”π’Šπ’β‘π’™+𝒄𝒐𝒔⁑𝒙 |γ€—+π‘ͺ

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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.