Example 7 - Find intgeral (i) cos2 x dx (ii) sin 2x cos 3x

Example 7 Find (i) ﷮﷮ cos﷮2﷯﷮𝑥﷯﷯ 𝑑𝑥 ﷮﷮ cos﷮2﷯﷮𝑥﷯﷯ 𝑑𝑥 = ﷮﷮ cos﷮2𝑥﷯ + 1﷮2﷯﷯﷯ 𝑑𝑥 = 1﷮2﷯ ﷮﷮ cos﷮2𝑥﷯+1﷯﷯ 𝑑𝑥 = 1﷮2﷯ ﷮﷮ cos﷮2𝑥﷯﷯𝑑𝑥+ ﷮﷮1﷯𝑑𝑥﷯ Thus, ﷮﷮ cos﷮2﷯﷮𝑥﷯﷯𝑑𝑥= 1﷮2﷯ ﷮﷮ cos﷮2𝑥﷯﷯𝑑𝑥+ ﷮﷮1﷯𝑑𝑥﷯ = 1﷮2﷯ 1﷮2﷯ sin﷮2𝑥﷯+𝐶1+𝑥+𝐶2﷯ = 1﷮2﷯ 1﷮2﷯ sin﷮2𝑥﷯+𝑥+ 𝐶1+𝐶2﷯﷯ = 1﷮4﷯ sin﷮2𝑥﷯+ 1﷮2﷯𝑥+𝐶 = 𝒙﷮𝟐﷯+ 𝟏﷮𝟒﷯ 𝐬𝐢𝐧﷮𝟐𝒙﷯+𝑪 Example 7 Find (ii) ﷮﷮ sin﷮2𝑥 ﷯ cos﷮3𝑥﷯﷯ 𝑑𝑥 Now ﷮﷮ sin﷮2𝑥 ﷯ cos﷮3𝑥﷯﷯﷯𝑑𝑥= 1﷮2﷯ ﷮﷮ sin﷮5𝑥﷯− sin﷮𝑥﷯﷯﷯𝑑𝑥 = 1﷮2﷯ ﷮﷮ sin﷮5𝑥﷯﷯𝑑𝑥− ﷮﷮ sin﷮𝑥﷯﷯𝑑𝑥﷯ Thus, ﷮﷮ sin﷮2𝑥 ﷯ cos﷮3𝑥﷯﷯= 1﷮2﷯ ﷮﷮ sin﷮5𝑥﷯𝑑𝑥﷯− ﷮﷮ sin﷮𝑥﷯﷯𝑑𝑥﷯ = 1﷮2﷯ − 1﷮5﷯ cos﷮5𝑥﷯+ 1﷮5﷯𝐶1− − cos﷮𝑥﷯﷯+𝐶2﷯ = 1﷮2﷯ − 1﷮5﷯ cos﷮5𝑥﷯+ cos﷮𝑥﷯+ 1﷮5﷯𝐶1−𝐶2﷯ = 1﷮2﷯ − 1﷮5﷯ cos﷮5𝑥﷯+ cos﷮𝑥﷯+𝐶﷯ = −𝟏﷮𝟏𝟎﷯ 𝐜𝐨𝐬﷮𝟓𝒙﷯+ 𝟏﷮𝟐﷯ 𝐜𝐨𝐬﷮𝒙﷯+𝑪 Example 7 Find (iii) ﷮﷮ sin﷮3﷯﷮𝑥﷯﷯ 𝑑𝑥 ﷮﷮ sin﷮3﷯﷮𝑥﷯﷯ 𝑑𝑥= ﷮﷮ 3 sin﷮𝑥﷯ − sin﷮3𝑥﷯﷮4﷯﷯ 𝑑𝑥 = 1﷮4﷯ ﷮﷮ 3 sin﷮𝑥﷯− sin﷮3𝑥﷯﷯﷯ 𝑑𝑥 = 1﷮4﷯ 3 ﷮﷮ sin﷮𝑥﷯﷯ 𝑑𝑥− ﷮﷮ sin﷮3𝑥﷯﷯ 𝑑𝑥﷯ Thus, ﷮﷮ sin﷮3﷯﷮𝑥﷯﷯𝑑𝑥= 1﷮4﷯ 3 ﷮﷮ sin﷮𝑥﷯𝑑𝑥﷯− ﷮﷮ sin﷮3𝑥﷯﷯𝑑𝑥﷯ = 1﷮4﷯ 3 − cos﷮𝑥﷯+𝐶2﷯− − 1﷮3﷯ cos﷮3𝑥﷯+ 1﷮3﷯𝐶1﷯﷯ = 1﷮4﷯ −3 cos﷮𝑥﷯+3 𝐶2+ 1﷮3﷯ cos﷮3𝑥﷯+ 1﷮3﷯𝐶1﷯ = 1﷮4﷯ −3 cos﷮𝑥﷯+ 1﷮3﷯ cos﷮3𝑥﷯+ 3 𝐶2− 1﷮3﷯𝐶1﷯﷯ =− 3﷮4﷯ cos﷮𝑥﷯+ 1﷮12﷯ cos﷮3𝑥﷯+ 1﷮4﷯ 3 𝐶2− 1﷮3﷯𝐶1﷯ =− 𝟑﷮𝟒﷯ 𝒄𝒐𝒔﷮𝒙﷯+ 𝟏﷮𝟏𝟐﷯ 𝒄𝒐𝒔﷮𝟑𝒙﷯+𝑪