Ex 7.6, 9 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.6, 9 𝑥 c𝑜 𝑠﷮−1﷯﷮𝑥﷯ ﷮﷮𝑥﷯ cos﷮−1﷯﷮𝑥 𝑑𝑥﷯ Let x = cos﷮𝜃﷯ dx = − sin﷮𝜃 𝑑𝜃﷯ ﷮﷮𝑥﷯ cos﷮−1﷯﷮𝑥 𝑑𝑥 ﷯ = ﷮﷮ cos﷮𝜃﷯﷯ 𝒄𝒐𝒔﷮−𝟏﷯( 𝒄𝒐𝒔﷮𝜽﷯)(− sin﷮𝜃 )𝑑𝜃﷯ = ﷮﷮ sin﷮𝜃 ﷯𝜽 cos﷮𝜃﷯𝑑𝜃﷯ = −1﷮2﷯ ﷮﷮ 𝜃 (2sin﷮𝜃 cos﷮𝜃) ﷯𝑑𝜃﷯﷯ = −1﷮2﷯ ﷮﷮ 𝜃 sin﷮2𝜃﷯𝑑𝜃﷯ = −1﷮2﷯ 𝜃∫ sin﷮2𝜃﷯𝑑𝜃−∫ 𝑑𝜃﷮𝑑𝜃﷯∫sin 2𝜃 𝑑𝜃﷯𝑑𝜃﷯ = −1﷮2﷯ 𝜃 −( cos﷮2𝜃﷯)﷮2﷯−∫1. − (cos﷮2𝜃﷯)﷮2﷯𝑑𝜃﷯ = −1﷮2﷯ −𝜃﷮2﷯ cos﷮2𝜃﷯+ 1﷮2﷯∫cos 2𝜃 𝑑𝜃﷯ = −1﷮2﷯ −𝜃﷮2﷯ cos﷮2𝜃﷯+ 1﷮2﷯ sin﷮2𝜃﷯﷮2﷯﷯ + C = 𝜃﷮4﷯ 𝒄𝒐𝒔﷮𝟐𝜽﷯ – 1﷮8﷯ 𝒔𝒊𝒏﷮𝟐𝜽﷯ + C = 𝜃﷮4﷯ ( 𝟐𝒄𝒐𝒔﷮𝟐﷯𝜽 – 1) – 1﷮8﷯ × 2 𝒔𝒊𝒏﷮𝜽﷯ 𝐜𝐨𝐬 𝜽 + C = 𝜃﷮4﷯ ( 2𝑐𝑜𝑠﷮2﷯𝜃 – 1) – 1﷮4﷯ 𝑠𝑖𝑛﷮𝜃﷯ 𝑐𝑜𝑠 𝜃 + C = 𝜃﷮4﷯ ( 2𝑐𝑜𝑠﷮2﷯𝜃 – 1) – 1﷮4﷯ 𝑐𝑜𝑠﷮𝜃﷯ ﷮1− cos﷮2﷯𝜃﷯ + C = cos﷮−1﷯﷮𝑥﷯﷮4﷯ 2 𝑥﷮2﷯−1﷯− 𝑥﷮4﷯ ﷮1− 𝑥﷮2﷯﷯+C = 𝟐𝒙﷮𝟐﷯ − 𝟏﷯﷮𝟒﷯ 𝒄𝒐𝒔﷮−𝟏﷯𝒙− 𝒙﷮𝟒﷯ ﷮𝟏− 𝒙﷮𝟐﷯﷯+𝐂