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Ex 7.11, 12 - Using properties, evaluate x dx / 1 + sin x dx

Ex 7.11, 12 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.11, 12 - Chapter 7 Class 12 Integrals - Part 3 Ex 7.11, 12 - Chapter 7 Class 12 Integrals - Part 4

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Ex 7.11, 12 By using the properties of definite integrals, evaluate the integrals: ∫_0^πœ‹β–’(π‘₯ 𝑑π‘₯)/(1 + sin⁑π‘₯ ) 𝑑π‘₯ Let I=∫_0^πœ‹β–’π‘₯/(1+ sin⁑π‘₯ ) 𝑑π‘₯ ∴ I=∫_0^πœ‹β–’(πœ‹ βˆ’ π‘₯)/(1+ sin⁑π‘₯ ) 𝑑π‘₯ Using P4 : ∫_0^π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯=γ€— ∫_0^π‘Žβ–’π‘“(π‘Žβˆ’π‘₯)𝑑π‘₯ Adding (1) and (2) i.e. (1) + (2) I+I=∫_0^πœ‹β–’( π‘₯)/(1 + sin⁑π‘₯ ) 𝑑π‘₯+∫_0^πœ‹β–’( πœ‹ βˆ’ π‘₯)/(1 + sin⁑π‘₯ ) 𝑑π‘₯ 2I=∫_0^πœ‹β–’( π‘₯ + πœ‹ βˆ’ π‘₯)/(1 + sin⁑π‘₯ ) 𝑑π‘₯ 2I=∫_0^πœ‹β–’( πœ‹)/(1 + sin⁑π‘₯ ) 𝑑π‘₯ I=πœ‹/2 ∫_0^πœ‹β–’( 1)/(1 + sin⁑π‘₯ ) 𝑑π‘₯ Multiplying and dividing by (1βˆ’sin⁑π‘₯ ) I=πœ‹/2 ∫_0^πœ‹β–’γ€–( 1)/(1 + sin⁑π‘₯ ) Γ— (1 βˆ’ sin⁑π‘₯)/(1 βˆ’ sin⁑π‘₯ )γ€— . 𝑑π‘₯ I=πœ‹/2 ∫_0^πœ‹β–’(1 βˆ’ sin⁑π‘₯)/(1 βˆ’ sin^2⁑π‘₯ ) 𝑑π‘₯ I=πœ‹/2 ∫_0^πœ‹β–’(1 βˆ’ sin⁑π‘₯)/( γ€–cos γ€—^2⁑π‘₯ ) 𝑑π‘₯ I=πœ‹/2 ∫_0^πœ‹β–’[1/cos^2⁑π‘₯ βˆ’sin⁑π‘₯/( γ€–cos γ€—^2⁑π‘₯ )] 𝑑π‘₯ I=πœ‹/2 ∫_0^πœ‹β–’[sec^2⁑π‘₯βˆ’sin⁑π‘₯/(cos⁑π‘₯ .γ€– cos〗⁑π‘₯ )] 𝑑π‘₯ I=πœ‹/2 ∫_0^πœ‹β–’[sec^2⁑π‘₯βˆ’tan⁑〖π‘₯ sec⁑π‘₯ γ€— ] 𝑑π‘₯ I=πœ‹/2 [[tan⁑π‘₯ ]_0^πœ‹βˆ’[sec⁑π‘₯ ]_0^πœ‹ ] I=πœ‹/2 [[π‘‘π‘Žπ‘›(πœ‹)βˆ’π‘‘π‘Žπ‘›(0)]βˆ’[𝑠𝑒𝑐(πœ‹)βˆ’π‘ π‘’π‘(0)]] I=πœ‹/2 [[0βˆ’0]βˆ’[βˆ’1βˆ’1]] I=πœ‹/2 [0βˆ’(βˆ’2)] I=πœ‹/2 [2] 𝐈=𝝅

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.