Ex 7.2, 37 - Integrate x3 sin (tan-1 x4) / 1 + x8 - Ex 7.2

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.2, 37 𝑥3 𝑠𝑖𝑛 ( tan﷮−1﷯﷮ 𝑥﷮4﷯﷯)﷮1+𝑥8﷯ Step 1: Let tan﷮−1﷯﷮ 𝑥﷮4﷯﷯= 𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 1﷮1 + 𝑥﷮4﷯﷯﷮2﷯﷯. 𝑑 𝑥﷮4﷯﷯﷮𝑑𝑥﷯= 𝑑𝑡﷮𝑑𝑥﷯ 1﷮1 + 𝑥﷮8﷯﷯. 4 𝑥﷮3﷯= 𝑑𝑡﷮𝑑𝑥﷯ 4 𝑥﷮3﷯﷮1 + 𝑥﷮8﷯﷯= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 1 + 𝑥﷮8﷯﷮4 𝑥﷮3﷯﷯ . 𝑑𝑡 Step 2: Integrating the function ﷮﷮ 𝑥3 𝑠𝑖𝑛 tan﷮−1﷯﷮ 𝑥﷮4﷯﷯﷯﷮1 + 𝑥8﷯﷯ . 𝑑𝑥 Putting 𝑡𝑎𝑛﷮−1﷯﷮ 𝑥﷮4﷯﷯=𝑡 & 𝑑𝑥= 1 + 𝑥﷮8﷯﷮4 𝑥﷮3﷯﷯ . 𝑑𝑡 = ﷮﷮ 𝑥3 𝑠𝑖𝑛 𝑡﷯﷮1 + 𝑥8﷯﷯. 1 + 𝑥﷮8﷯﷮4 𝑥﷮3﷯﷯𝑑𝑡 = ﷮﷮ sin﷮𝑡﷯﷮4﷯﷯𝑑𝑡 = 1﷮4﷯ ﷮﷮ sin﷮𝑡﷯﷯. 𝑑𝑡 = − 1﷮4﷯ cos﷮𝑡﷯+ 𝐶 = − 𝟏﷮𝟒﷯ 𝒄𝒐𝒔﷮ 𝐭𝐚𝐧﷮−𝟏﷯﷮ 𝒙﷮𝟒﷯﷯﷯﷯+𝑪

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