Check sibling questions

Misc 22 - Chapter 7 Class 12 Integrals

Last updated at May 29, 2023 by

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

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

Book a free demo

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

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

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.