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Ex 7.10, 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‘₯ 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

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