Misc 22 - Chapter 7 Class 12 Integrals

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