Integration by partial fraction - Type 1

Chapter 7 Class 12 Integrals
Concept wise

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class

### Transcript

Misc 22 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.