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Misc 23 - Integrate tan-1 root  (1 - x) / (1 + x) - Integration by partial fraction - Type 1

Misc 23 - Chapter 7 Class 12 Integrals - Part 2
Misc 23 - Chapter 7 Class 12 Integrals - Part 3

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Transcript

Misc 23 Integrate the function tan﷮−1﷯﷮ ﷮ 1 − 𝑥﷮1 + 𝑥﷯﷯﷯ Let x = cos 2𝜃 𝑑𝑥﷮𝑑𝜃﷯=−2 sin﷮2𝜃 ﷯ dx = −2 sin 2𝜃 d𝜃 Substituting, ﷮﷮ tan﷮−1﷯﷮ ﷮ 1 − 𝑥﷮1 + 𝑥﷯﷯﷯ 𝑑𝑥﷯ = ﷮﷮ 𝑡𝑎𝑛﷮−1﷯﷯ ﷮ 1 − cos﷮2𝜃﷯﷮1 + cos﷮2𝜃﷯﷯﷯×(−2 sin﷮2 𝜃)﷯ 𝑑 𝜃 = −2 ﷮﷮ 𝑡𝑎𝑛﷮−1﷯﷯ ﷮ 1 − 1 − 2 𝑠𝑖𝑛﷮2﷯ 𝜃﷯﷮1 + 2 𝑐𝑜𝑠﷮2﷯ 𝜃 − 1﷯﷯﷯× sin 2𝜃 d𝜃 ﷯ = −2 ﷮﷮ 𝑡𝑎𝑛﷮−1﷯﷯ ﷮ sin﷮2﷯﷮𝜃﷯﷮ cos﷮2﷯﷮𝜃﷯﷯﷯﷯× sin﷮2𝜃 𝑑𝜃﷯ = −2 ﷮﷮ 𝑡𝑎𝑛﷮−1﷯﷯ sin﷮𝜃﷯﷮ cos﷮𝜃﷯﷯﷯× sin﷮2𝜃 𝑑𝜃﷯ = −2 ﷮﷮ 𝑡𝑎𝑛﷮−1﷯﷯ 𝑡𝑎𝑛﷮𝜃﷯﷯× sin﷮2𝜃 𝑑𝜃﷯ = − 2 ﷮﷮𝜃﷯ sin﷮2𝜃 𝑑𝜃﷯ =−2 𝜃 ﷮﷮ sin﷮2𝜃﷯ 𝑑𝜃− ﷮﷮ 𝑑 𝜃﷯﷮𝑑𝜃﷯﷯ ﷮﷮ sin﷮2𝜃﷯﷯ 𝑑𝜃﷯ 𝑑𝜃﷯﷯ =−2 𝜃 − cos﷮2𝜃﷯﷮2﷯﷯− ﷮﷮1 − cos﷮2𝜃﷯﷮2﷯﷯﷯𝑑𝜃﷯ =−2 −𝜃 cos﷮2𝜃﷯﷮2﷯﷯+ ﷮﷮ cos﷮2𝜃﷯﷮2﷯﷯𝑑𝜃﷯ =−2 − 𝜃 cos﷮2𝜃﷯﷮2﷯+ sin﷮2𝜃﷯﷮4﷯﷯ Now, x = cos 2𝜃 Putting the values = −2 − 1﷮2﷯ 1﷮2﷯ 𝑐𝑜𝑠﷮−1﷯𝑥﷯𝑥+ ﷮1 − 𝑥﷮2﷯﷯﷮4﷯﷯ = −2 ﷮1 − 𝑥﷮2﷯﷯﷮4﷯− 𝑥 𝑐𝑜𝑠﷮−1﷯𝑥﷮4﷯﷯+ C = 𝟏﷮𝟐﷯ 𝒙 𝒄𝒐𝒔﷮−𝟏﷯𝒙− ﷮𝟏− 𝒙﷮𝟐﷯﷯ ﷯+ C

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.