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Misc 25 - Evaluate definite integral ex (1-sin x / 1-cos x) - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 25 Evaluate the definite integral ﷐﷐𝜋﷮2﷯﷮𝜋﷮﷐﷐e﷮𝑥﷯﷮﷐﷐1 −﷐sin﷮𝑥﷯﷮1 −﷐cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥﷯ ﷐﷐𝜋﷮2﷯﷮𝜋﷮﷐﷐e﷮𝑥﷯﷮﷐﷐1 − ﷐sin﷮𝑥﷯﷮1 −﷐ cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥﷯ = ﷐﷐𝜋﷮2﷯﷮𝜋﷮﷐﷐e﷮𝑥﷯﷮﷐﷐1 ﷮1 − ﷐cos﷮𝑥﷯﷯−﷐﷐sin﷮𝑥﷯﷮1 − ﷐cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥﷯ Let f(x) = ﷐−﷐sin﷮𝑥﷯﷮1 − ﷐cos﷮𝑥﷯﷯ f’(x) = −﷐﷐𝑐𝑜𝑠𝑥 (1 − ﷐cos﷮𝑥) − (﷐sin﷮𝑥) (﷐sin﷮𝑥)﷯﷯﷯﷮(1 − ﷐﷐cos﷮𝑥﷯)﷮2﷯﷯﷯ =−﷐﷐﷐cos﷮𝑥 −﷯ ﷐𝑐𝑜𝑠﷮2﷯𝑥 − ﷐𝑠𝑖𝑛﷮2﷯𝑥﷮(1 − ﷐cos 𝑥⁡)﷮2﷯﷯﷯ = −﷐﷐﷐cos﷮𝑥 − ﷐﷐𝑐𝑜𝑠﷮2﷯𝑥 + ﷐𝑠𝑖𝑛﷮2﷯𝑥﷯﷯﷮(1 − ﷐𝑐𝑜𝑠 𝑥⁡)﷮2﷯﷯﷯ = ﷐﷐﷐cos﷮𝑥 − 1﷯﷮﷐(1 −﷐cos﷮𝑥)﷯﷮2﷯﷯﷯ = ﷐− 1 ﷐cos﷮𝑥 ﷯﷮﷐(1 −﷐cos﷮𝑥)﷯﷮2﷯﷯ = ﷐1﷮1 − ﷐cos﷮𝑥﷯﷯ Hence the given integration is of form, ﷐﷮﷮﷐𝑒﷮𝑥﷯(𝑓﷐𝑥﷯+﷐𝑓﷮′﷯﷐𝑥﷯) 𝑑𝑥﷯= ﷐𝑒﷮𝑥﷯𝑓(𝑥) Where f(x) = ﷐−﷐sin﷮𝑥﷯﷮1 −﷐cos﷮𝑥﷯﷯ f’(x) = ﷐1﷮1 − ﷐cos﷮𝑥﷯﷯ Hence, ﷐﷐𝜋﷮2﷯﷮𝜋﷮﷐﷐e﷮𝑥﷯﷮﷐﷐1 − ﷐sin﷮𝑥﷯﷮1 −﷐ cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥﷯ = ﷐﷐𝜋﷮2﷯﷮𝜋﷮﷐﷐e﷮𝑥﷯﷮﷐﷐1 ﷮1 − ﷐cos﷮𝑥﷯﷯−﷐﷐sin﷮𝑥﷯﷮1 − ﷐cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥﷯ = ﷐﷐﷐𝑒﷮𝑥﷯ 𝑓(𝑥)﷯﷮﷐𝜋﷮2﷯﷮𝜋﷯ = ﷐﷐﷐𝑒﷮𝑥﷯ ﷐﷐−﷐sin﷮𝑥﷯﷮1 −﷐cos﷮𝑥﷯﷯﷯﷯﷮﷐𝜋﷮2﷯﷮𝜋﷯ = ﷐﷐﷐﷐𝑒﷮𝑥﷯﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥 −1﷯﷯﷯﷮﷐𝜋﷮2﷯﷮𝜋﷯ Putting limits, = ﷐﷐𝑒﷮𝜋﷯﷐sin﷮𝜋﷯﷮﷐cos﷮𝜋 −1﷯﷯− ﷐﷐𝑒﷮﷐𝜋﷮2﷯﷯ ﷐sin﷮﷐𝜋﷮2﷯﷯﷮﷐cos﷮ ﷐𝜋﷮2﷯ −1﷯﷯ = ﷐﷐𝑒﷮𝜋﷯(0)﷮−1 −1﷯− ﷐﷐𝑒﷮﷐𝜋﷮2﷯﷯ (1)﷮0 − 1﷯ = ﷐𝒆﷮﷐𝝅﷮𝟐﷯﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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