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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise

Transcript

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
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.