Ex 7.9, 8 - Direct Integrate cosec x dx from pi/6 to pi/4 - Ex 7.9

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex7.9, 8 ๐œ‹๏ทฎ6๏ทฏ๏ทฎ ๐œ‹๏ทฎ4๏ทฏ๏ทฎ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ๏ทฎ๐‘‘๐‘ฅ๏ทฏ๏ทฏ Let F ๐‘ฅ๏ทฏ= ๏ทฎ๏ทฎ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ . ๐‘‘๐‘ฅ๏ทฏ Divide and multiply by ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ F ๐‘ฅ๏ทฏ= ๏ทฎ๏ทฎ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ๏ทฎ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ๏ทฏ ๐‘‘๐‘ฅ Let c๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ=๐‘ก Differentiating w.r.t. ๐‘ฅ ๐‘‘๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ= ๐‘‘๐‘ก๏ทฎ๐‘‘๐‘ฅ๏ทฏ โˆ’๐‘๐‘œ๐‘ ๐‘’ ๐‘๏ทฎ2๏ทฏ ๐‘ฅโˆ’๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ ๐‘๐‘œ๐‘ก ๐‘ฅ= ๐‘‘๐‘ก๏ทฎ๐‘‘๐‘ฅ๏ทฏ โˆ’๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ= ๐‘‘๐‘ก๏ทฎ๐‘‘๐‘ฅ๏ทฏ ๐‘‘๐‘ฅ= ๐‘‘๐‘ก๏ทฎโˆ’๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ +๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ๏ทฏ Therefore, ๏ทฎ๏ทฎ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ๏ทฎ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ๏ทฏ ๐‘‘๐‘ฅ = ๏ทฎ๏ทฎ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ๏ทฎ๐‘ก๏ทฏ๏ทฏ. ๐‘‘๐‘ก๏ทฎโˆ’๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ + ๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ๏ทฏ =โˆ’ ๏ทฎ๏ทฎ ๐‘‘๐‘ก๏ทฎ๐‘ก๏ทฏ๏ทฏ =โˆ’ log๏ทฎ ๐‘ก๏ทฏ๏ทฏ =โˆ’ log๏ทฎ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+๐‘๐‘œ๐‘ก ๐‘ฅ๏ทฏ๏ทฏ Hence, F ๐‘ฅ๏ทฏ=โˆ’ log๏ทฎ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ+ cot๏ทฎ๐‘ฅ๏ทฏ๏ทฏ๏ทฏ Step 2 :- ๐œ‹๏ทฎ6๏ทฏ๏ทฎ ๐œ‹๏ทฎ4๏ทฏ๏ทฎ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐‘ฅ=๐น ๐œ‹๏ทฎ4๏ทฏ๏ทฏโˆ’๐น ๐œ‹๏ทฎ6๏ทฏ๏ทฏ๏ทฏ =โˆ’๐‘™๐‘œ๐‘” ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐œ‹๏ทฎ4๏ทฏ๏ทฏ+๐‘๐‘œ๐‘ก ๐œ‹๏ทฎ4๏ทฏ๏ทฏ๏ทฏโˆ’ โˆ’ log๏ทฎ ๐‘๐‘œ๐‘ ๐‘’๐‘ ๐œ‹๏ทฎ6๏ทฏ๏ทฏ+ ๐‘๐‘œ๐‘ก ๐œ‹๏ทฎ6๏ทฏ๏ทฏ๏ทฏ๏ทฏ๏ทฏ =โˆ’๐‘™๐‘œ๐‘” ๏ทฎ2๏ทฏ+1๏ทฏ+๐‘™๐‘œ๐‘” 2+ ๏ทฎ3๏ทฏ๏ทฏ =โˆ’๐‘™๐‘œ๐‘” 2+ ๏ทฎ3๏ทฏ๏ทฏโˆ’ ๐‘™๐‘œ๐‘” ๏ทฎ2๏ทฏ+1๏ทฏ =๐‘™๐‘œ๐‘” 2 + ๏ทฎ3๏ทฏ๏ทฎ ๏ทฎ2๏ทฏ + 1๏ทฏ๏ทฏ =๐‘™๐‘œ๐‘” 2 + ๏ทฎ3๏ทฏ๏ทฎ ๏ทฎ2๏ทฏ + 1๏ทฏร— 2 โˆ’ ๏ทฎ3๏ทฏ๏ทฎ2 โˆ’ ๏ทฎ3๏ทฏ๏ทฏ๏ทฏ =๐‘™๐‘œ๐‘” 2๏ทฏ๏ทฎ2๏ทฏโˆ’ ๏ทฎ3๏ทฏ๏ทฏ๏ทฎ2๏ทฏ๏ทฎ ๏ทฎ2๏ทฏ+1๏ทฏ ร— 2โˆ’ ๏ทฎ3๏ทฏ๏ทฏ๏ทฏ๏ทฏ =๐‘™๐‘œ๐‘” 4 โˆ’ 3๏ทฎ ๏ทฎ2๏ทฏ+1๏ทฏ 2โˆ’ ๏ทฎ3๏ทฏ๏ทฏ๏ทฏ๏ทฏ =๐‘™๐‘œ๐‘” 1 ร— ๏ทฎ2๏ทฏ โˆ’ 1๏ทฏ๏ทฎ 2โˆ’ ๏ทฎ3๏ทฏ๏ทฏ ๏ทฎ2๏ทฏ+1๏ทฏ ร— ๏ทฎ2๏ทฏโˆ’1๏ทฏ๏ทฏ๏ทฏ =๐‘™๐‘œ๐‘” ๏ทฎ2๏ทฏ โˆ’ 1๏ทฎ 2โˆ’ ๏ทฎ3๏ทฏ๏ทฏ ๏ทฎ2๏ทฏ๏ทฎ2๏ทฏโˆ’ 1๏ทฎ2๏ทฏ๏ทฏ๏ทฏ๏ทฏ =๐’๐’๐’ˆ ๏ทฎ๐Ÿ๏ทฏ โˆ’ ๐Ÿ๏ทฎ๐Ÿ โˆ’ ๏ทฎ๐Ÿ‘๏ทฏ๏ทฏ๏ทฏ

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