Example 5 - Integrate (i) sin mx (ii) 2x sin (x2 + 1) - Integration by substitution - Trignometric - Normal

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 5 Integrate the following functions 𝑤.𝑟.𝑡. 𝑥: (i) sin⁡𝑚𝑥 ﷮﷮ sin﷮ 𝑚𝑥﷯ 𝑑𝑥﷯﷯ Let t = mx 𝑑𝑡﷮𝑑𝑥﷯=𝑚 𝑑𝑡﷮𝑚﷯=𝑑𝑥 Substituting values ﷮﷮ sin﷮ 𝑚𝑥﷯ 𝑑𝑥﷯﷯ = ﷮﷮𝑠𝑖𝑛𝑡× 𝑑𝑡﷮𝑚﷯﷯ = 1﷮𝑚﷯ ﷮﷮ sin﷮𝑡 𝑑𝑡﷯﷯ = 1﷮𝑚﷯ ﷮﷮ sin﷮𝑡 𝑑𝑡﷯﷯ = 1﷮𝑚﷯ − cos﷮𝑡﷯﷯+ C = − cos﷮𝑡﷯﷮𝑚﷯+ C Putting back value of t = m𝑥 = − 𝐜𝐨𝐬﷮𝒎𝒙﷯﷮𝒎﷯+ C Example 5 Integrate the following functions 𝑤.𝑟.𝑡. 𝑥: (ii) 2𝑥 sin 𝑥﷮2﷯+1﷯ ﷮﷮2𝑥 sin 𝑥﷮2﷯+1﷯﷯ dx Let t = 𝑥﷮2﷯+1 𝑑𝑡﷮𝑑𝑥﷯=2𝑥 dt = 2x dx Substituting ﷮﷮2𝑥 sin﷮ 𝑥﷮2﷯+1﷯ 𝑑𝑥﷯﷯ = ﷮﷮ sin﷮𝑡 𝑑𝑡﷯﷯ = − cos﷮𝑡﷯﷯+ C Putting value of t = 𝑥﷮2﷯+1 = − cos 𝒙﷮𝟐﷯+𝟏﷯+ C Example 5 Integrate the following functions 𝑤.𝑟.𝑡. 𝑥: (iii) tan﷮4﷯﷮ ﷮𝑥﷯﷯ sec﷮2﷯﷮ ﷮𝑥﷯﷯﷮ ﷮𝑥﷯﷯ ﷮﷮ tan﷮4﷯﷮ ﷮𝑥﷯﷯ sec﷮2﷯﷮ ﷮𝑥﷯﷯﷮ ﷮𝑥﷯﷯﷯ Let t = tan ﷮𝑥﷯ 𝑑𝑡﷮𝑑𝑥﷯= 𝑠𝑒𝑐﷮2﷯ ﷮𝑥﷯× 1﷮2 ﷮𝑥 ﷯﷯ 2dt = dx 𝑠𝑒𝑐﷮2﷯ ﷮𝑥﷯ Substituting value, ﷮﷮ 𝑡𝑎𝑛﷮4﷯ ﷮𝑥﷯× 𝑠𝑒𝑐﷮2﷯ ﷮𝑥﷯ 𝑑𝑥﷮ ﷮𝑥﷯﷯﷯ = ﷮﷮ 𝑡﷮4﷯﷯×2 𝑑𝑡 = 2 ﷮﷮ 𝑡﷮4﷯ 𝑑𝑡﷯ = 2 𝑡﷮5﷯﷮5﷯﷯+ C Putting back value of t = 2 𝑡𝑎𝑛﷮5﷯ ﷮𝑥﷯﷯﷮5﷯+ C Example 5 Integrate the following functions 𝑤.𝑟.𝑡. 𝑥: (iv) sin﷮ tan﷮−1﷯﷮𝑥﷯﷯﷯﷮1 + 𝑥﷮2﷯﷯ sin ﷮ tan﷮−1﷯﷮𝑥﷯﷯﷯﷮1 + 𝑥﷮2﷯﷯ Let t = 𝑡𝑎𝑛﷮−1﷯ 𝑥 𝑑𝑡﷮𝑑𝑥﷯= 1﷮1+ 𝑥﷮2﷯﷯ 𝑑𝑡 = 𝑑𝑥﷮1+ 𝑥﷮2﷯﷯ Substituting ﷮﷮ sin ﷮ tan﷮−1﷯﷮𝑥﷯﷯﷯﷮1 + 𝑥﷮2﷯﷯ ﷯= ﷮﷮ sin﷮𝑡 𝑑𝑡﷯﷯ = − cos t + C Putting value of t = 𝑡𝑎𝑛﷮−1﷯ 𝑥 = − cos (𝑡𝑎𝑛﷮−1﷯𝑥)+ C

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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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