Misc 32 - Definite integral x tan x / sec x + tanx - Definate Integration by properties - P4

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 32 Evaluate the definite integral ﷐0﷮𝜋﷮﷐𝑥﷐tan﷮𝑥﷯ ﷮﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯﷯﷯ 𝑑𝑥 Let I=﷐0﷮𝜋﷮﷐𝑥﷐tan﷮𝑥﷯ ﷮﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯﷯﷯ 𝑑𝑥 ∴ I=﷐0﷮𝜋﷮﷐﷐𝜋 − 𝑥﷯﷐tan﷮ ﷐𝜋 − 𝑥﷯﷯﷮﷐sec﷮﷐𝜋 − 𝑥﷯﷯ +﷐ tan﷮﷐𝜋 − 𝑥﷯﷯﷯﷯ 𝑑𝑥 I=﷐0﷮𝜋﷮﷐﷐𝜋 − 𝑥﷯(−﷐tan﷮ 𝑥﷯) ﷮(−﷐sec﷮ 𝑥﷯) + ﷐( −tan﷮𝑥﷯)﷯﷯ 𝑑𝑥 I=﷐0﷮𝜋﷮﷐−﷐𝜋 − 𝑥﷯﷐tan﷮𝑥﷯ ﷮−(﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯)﷯﷯ 𝑑𝑥 I =﷐0﷮𝜋﷮﷐﷐𝜋 − 𝑥﷯﷐(−tan﷮𝑥)﷯ ﷮(−﷐sec﷮𝑥)﷯ +(−)﷐ (tan﷮𝑥)﷯﷯﷯ 𝑑𝑥 Adding (1) and (2) i.e. (1) + (2) I+I=﷐0﷮𝜋﷮﷐𝑥﷐tan﷮𝑥﷯ ﷮﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯﷯﷯ 𝑑𝑥+﷐0﷮𝜋﷮﷐𝜋﷐tan﷮𝑥﷯ − 𝑥﷐tan﷮𝑥﷯﷮﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯﷯﷯ 𝑑𝑥 2I=﷐0﷮𝜋﷮﷐𝑥﷐tan﷮𝑥﷯ + 𝜋﷐tan﷮𝑥﷯ − 𝑥﷐tan﷮𝑥﷯﷮﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯﷯﷯ 𝑑𝑥 2I=﷐0﷮𝜋﷮﷐𝜋﷐tan﷮𝑥﷯﷮﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯﷯﷯ 𝑑𝑥 2I=𝜋﷐0﷮𝜋﷮﷐﷐tan﷮𝑥﷯﷮﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯﷯﷯ 𝑑𝑥 I=﷐𝜋﷮2﷯ ﷐0﷮𝜋﷮﷐﷐tan﷮𝑥﷯﷮﷐sec﷮𝑥﷯ +﷐ tan﷮𝑥﷯﷯﷯ 𝑑𝑥 =﷐𝜋﷮2﷯ ﷐0﷮𝜋﷮﷐﷐﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥﷯﷯﷮﷐1﷮﷐cos﷮𝑥﷯﷯ + ﷐﷐sin﷮𝑥﷯﷮﷐cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 =﷐𝜋﷮2﷯ ﷐0﷮𝜋﷮﷐﷐sin﷮𝑥﷯﷮1 + ﷐sin﷮𝑥﷯﷯﷯ 𝑑𝑥 =﷐𝜋﷮2﷯ ﷐0﷮𝜋﷮﷐﷐sin﷮𝑥﷯ + 1 − 1﷮1 + ﷐sin﷮𝑥﷯﷯﷯ 𝑑𝑥 =﷐𝜋﷮2﷯ ﷐0﷮𝜋﷮﷐﷐1 + ﷐sin﷮𝑥﷯﷮1 + ﷐sin﷮𝑥﷯﷯ −﷐1﷮1 + ﷐sin﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 =﷐𝜋﷮2﷯ ﷐0﷮𝜋﷮﷐1 −﷐1﷮1 + ﷐sin﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 =﷐𝜋﷮2﷯ ﷐﷐0﷮𝜋﷮1﷯ 𝑑𝑥−﷐0﷮𝜋﷮﷐1﷮1 + ﷐sin﷮𝑥﷯﷯﷯ 𝑑𝑥﷯ =﷐𝜋﷮2﷯ ﷐﷐﷐𝑥﷯﷮0﷮𝜋﷯−﷐0﷮𝜋﷮﷐1﷮1 + ﷐sin﷮𝑥﷯﷯﷯ ﷐﷐1 − ﷐sin﷮𝑥﷯﷮1 − ﷐sin﷮𝑥﷯﷯﷯ 𝑑𝑥﷯ =﷐𝜋﷮2﷯ ﷐﷐𝜋−0﷯−﷐0﷮𝜋﷮﷐1 − ﷐sin﷮𝑥﷯﷮1 − ﷐﷐sin﷮2﷯﷮𝑥﷯﷯﷯ 𝑑𝑥﷯ =﷐𝜋﷮2﷯ ﷐𝜋−﷐0﷮𝜋﷮﷐1 − ﷐sin﷮𝑥﷯﷮﷐﷐cos﷮2﷯﷮𝑥﷯﷯﷯ 𝑑𝑥﷯ =﷐𝜋﷮2﷯ ﷐𝜋−﷐0﷮𝜋﷮﷐﷐1﷮﷐﷐cos﷮2﷯﷮𝑥﷯﷯ − ﷐﷐sin﷮𝑥﷯﷮﷐﷐cos﷮2﷯﷮𝑥﷯﷯﷯﷯ 𝑑𝑥﷯ =﷐𝜋﷮2﷯ ﷐𝜋−﷐0﷮𝜋﷮﷐﷐﷐sec﷮2﷯﷮𝑥﷯−﷐tan﷮𝑥﷯﷐sec﷮𝑥﷯﷯﷯𝑑𝑥﷯ =﷐𝜋﷮2﷯ ﷐𝜋−﷐0﷮𝜋﷮﷐﷐sec﷮2﷯﷮𝑥﷯﷯𝑑𝑥+﷐0﷮𝜋﷮﷐tan﷮𝑥﷯﷐sec﷮𝑥﷯﷯𝑑𝑥﷯ =﷐𝜋﷮2﷯ ﷐𝜋−﷐﷐﷐tan﷮𝑥﷯﷯﷮0﷮𝜋﷯+﷐﷐﷐sec﷮𝑥﷯﷯﷮0﷮𝜋﷯﷯ =﷐𝜋﷮2﷯﷐𝜋−﷐﷐tan﷮﷐𝜋﷯−﷐tan﷮﷐0﷯﷯﷯﷯+﷐sec ﷐𝜋﷯−﷐sec﷮﷐0﷯﷯﷯﷯ =﷐𝜋﷮2﷯﷐𝜋−﷐0−0﷯+﷐−1−1﷯﷯ =﷐𝜋﷮2﷯﷐𝜋−0+﷐−2﷯﷯ =﷐𝝅﷮𝟐﷯﷐𝝅−𝟐﷯

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