Misc 20

Transcript

Misc 20 Integrate the function ﷮ 1 − ﷮𝑥﷯﷮1 + ﷮𝑥﷯﷯﷯ ﷮﷮ ﷮ 1 − ﷮𝑥﷯﷮1 + ﷮𝑥﷯﷯﷯ 𝑑𝑥﷯ Let x = 𝑐𝑜𝑠﷮2﷯ 2𝜃 dx = −4 cos 2𝜃 sin 2𝜃 d𝜃 Substituting, = ﷮﷮ ﷮ 1 − ﷮ 𝑐𝑜𝑠﷮2﷯ 2𝜃﷯﷯﷮1 + ﷮ 𝑐𝑜𝑠﷮2﷯ 2𝜃﷯﷯﷯﷯﷯×−4 cos﷮2θ﷯ sin﷮2θ﷯ 𝑑𝜃 = ﷮﷮ ﷮ 1 − cos﷮2𝜃﷯﷮1 + 𝑐𝑜𝑠2 𝜃﷯﷯﷯× −4﷯ cos﷮2θ﷯ sin﷮2θ﷯ 𝑑𝜃 = −4 ﷮﷮ ﷮ 1 − 1 − 2 𝑠𝑖𝑛﷮2 ﷯𝜃﷯﷮1 + 2 𝑐𝑜𝑠﷮2 ﷯𝜃 − 1﷯﷯﷯﷯ cos﷮2θ﷯ (2 sin﷮θ﷯ cos﷮𝜃)﷯ 𝑑 𝜃 = −8 ﷮﷮ ﷮ 2 𝑠𝑖𝑛﷮2 ﷯𝜃﷮2 𝑐𝑜𝑠﷮2 ﷯𝜃﷯﷯﷯ cos﷮2θ﷯ cos﷮𝜃﷯ sin﷮𝜃﷯𝑑 𝜃 = −8 ﷮﷮ sin﷮𝜃﷯﷮ cos﷮θ﷯﷯﷯ cos﷮2θ﷯ cos﷮𝜃﷯ sin﷮𝜃﷯𝑑 𝜃 = −8 ﷮﷮ 𝑠𝑖𝑛﷮2﷯ 𝜃﷯ cos﷮2θ﷯ 𝑑 𝜃 = −8 ﷮﷮ 1 − cos﷮2θ﷯﷮2﷯﷯﷯ cos﷮2θ﷯ 𝑑 𝜃 = + 4 ﷮﷮ 𝑐𝑜𝑠﷮2﷯ 2θ− cos﷮2θ﷯﷯﷯ 𝑑 𝜃 = + 4 ﷮﷮ 𝑐𝑜𝑠﷮2﷯ 2 𝜃−𝑐𝑜𝑠2 𝜃﷯﷯ 𝑑 𝜃 = 4 ﷮﷮ cos﷮4 𝜃 + 1﷯﷮2﷯﷯ 𝑑 𝜃 − 4 ﷮﷮𝑐𝑜𝑠2 𝜃﷯ 𝑑 𝜃 = 2 sin﷮4﷯ 𝜃﷮4﷯+𝜃﷯ − 4 sin﷮2﷯ 𝜃﷮2﷯﷯+C = sin﷮4 𝜃﷯﷮2﷯+2 𝜃 − 2 𝑠𝑖𝑛2 𝜃+ C Now x = 𝑐𝑜𝑠﷮2﷯2 𝜃 1﷮2﷯ 𝑐𝑜𝑠﷮−1﷯ ﷮𝑥﷯=𝜃 1 − 𝑥= 𝑠𝑖𝑛﷮2﷯2 𝜃 sin 2𝜃 = ﷮1−𝑥﷯ Also, sin 4𝜃 = 2 sin 2𝜃 cos 2𝜃 = 2 ﷮1−𝑥﷯× ﷮𝑥﷯ = 2 ﷮𝑥﷯ ﷮1−𝑥﷯ Putting the values. = sin﷮4𝜃﷯﷮2﷯+2θ−2 sin﷮2θ﷯+ C = 2 ﷮𝑥﷯ ﷮1 − 𝑥﷯﷮2﷯+2 𝑐𝑜𝑠﷮−1﷯ ﷮𝑥﷯﷮2﷯−2 ﷮1−𝑥﷯+C = ﷮𝑥﷯ ﷮1−𝑥﷯+ 𝑐𝑜𝑠﷮−1﷯ ﷮𝑥﷯−2 ﷮1−𝑥﷯+ C = ﷮𝑥− 𝑥﷮2﷯﷯+ 𝑐𝑜𝑠﷮−1﷯ ﷮𝑥﷯−2 ﷮1−𝑥﷯+C = –2 ﷮𝟏−𝒙﷯+ 𝒄𝒐𝒔﷮−𝟏﷯ ﷮𝒙﷯+ ﷮𝒙− 𝒙﷮𝟐﷯﷯+𝐂