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Ex 7.11, 20 - Value of (x3 + x cos x + tan5 x + 1) dx - Definate Integration by properties - P7

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.11, 20 Choose the correct answer : The value of − 𝜋﷮2﷯﷮ 𝜋﷮2﷯﷮ 𝑥﷮3﷯+𝑥 𝑐𝑜𝑠 𝑥+ tan﷮5﷯﷮𝑥+1﷯﷯ 𝑑𝑥﷯ (A) 0 (B) 2 (C) 𝜋 (D) 1 − 𝜋﷮2﷯﷮ 𝜋﷮2﷯﷮ 𝑥﷮3﷯+𝑥 𝑐𝑜𝑠 𝑥+ tan﷮5﷯﷮𝑥+1﷯﷯ 𝑑𝑥﷯ −𝑎﷮+𝑎﷮𝑓 𝑥﷯ 𝑑𝑥﷯= 0, 𝑖𝑓 𝑓 −𝑥﷯=−𝑓 𝑥﷯﷮&2 0﷮𝑎﷮𝑓 𝑥﷯ 𝑑𝑥﷯, 𝑖𝑓 𝑓 −𝑥﷯=−𝑓 𝑥﷯﷯﷯ Hence = − 𝜋﷮2﷯﷮ 𝜋﷮2﷯﷮ 𝑥﷮3﷯+𝑥 𝑐𝑜𝑠 𝑥+ tan﷮5﷯﷮𝑥+1﷯﷯ 𝑑𝑥﷯ = 0﷮ 𝜋﷮2﷯﷮0 𝑑𝑥+ 0﷮ 𝜋﷮2﷯﷮0 𝑑𝑥+ 0﷮ 𝜋﷮2﷯﷮0 𝑑𝑥+2 0﷮ 𝜋﷮2﷯﷮𝑑𝑥﷯﷯﷯﷯ = 2 0﷮ 𝜋﷮2﷯﷮𝑑𝑥﷯ = 2 𝑥﷯﷮0﷮ 𝜋﷮2﷯﷯ = 2 𝜋﷮2﷯﷯=𝜋 Now taking I2 i.e. I2 = −𝜋﷮2﷯﷮ 𝜋﷮2﷯﷮1.𝑑𝑥﷯ I2 = 𝑥﷯﷮ −𝜋﷮2﷯﷮ 𝜋﷮2﷯﷯ I2 = 𝜋﷮2﷯− −𝜋﷮2﷯﷯ I2 = 𝜋﷮2﷯+ 𝜋﷮2﷯ I2 = 𝜋 + 𝜋﷮2﷯ I2 = 2𝜋﷮2﷯ I2 = 𝜋 Put the value of I1 and I2 in equation (1), we get −𝜋﷮2﷯﷮ 𝜋﷮2﷯﷮ 𝑥﷮3﷯+𝑥𝑐𝑜𝑠𝑥+ tan﷮𝑥+1﷯﷯𝑑𝑥﷯ = −𝜋﷮2﷯﷮ 𝜋﷮2﷯﷮ 𝑥﷮3﷯+𝑥𝑐𝑜𝑠 𝑥+ tan﷮𝑥﷯﷯𝑑𝑥+﷯ −𝜋﷮2﷯﷮ 𝜋﷮2﷯﷮1.𝑑𝑥﷯ = 0 + π = π ∴ Option C

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