Ex 7.2, 24 - Integrate (2 cosx - 3 sinx) / (6cos x + 4sin x)

Ex 7.2, 24 2 cos﷮𝑥 − 3 sin﷮𝑥﷯﷯﷮6 cos﷮𝑥 + 4 sin﷮𝑥﷯﷯﷯ Step 1: Let 6 cos﷮𝑥 + 4 sin﷮𝑥﷯﷯=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 −6 sin﷮𝑥 + 4 co𝑠﷮𝑥﷯﷯= 𝑑𝑡﷮𝑑𝑥﷯ 4 cos﷮𝑥﷯−6 sin﷮𝑥﷯﷯𝑑𝑥 =𝑑𝑡 𝑑𝑥 = 𝑑𝑡﷮4 cos﷮𝑥﷯ − 6 sin﷮𝑥﷯﷯ Step 2: Integrating the function ﷮﷮ 2 cos﷮𝑥 − 3 sin﷮𝑥﷯﷯﷮6 cos﷮𝑥 + 4 sin﷮𝑥﷯﷯﷯ ﷯. 𝑑𝑥 Putting 6 𝑐𝑜𝑠﷮𝑥+4 𝑠𝑖𝑛﷮𝑥﷯﷯=𝑡 & 𝑑𝑥= 𝑑𝑡﷮4 cos﷮𝑥﷯ − 6 sin﷮𝑥﷯﷯ = ﷮﷮ 2 cos﷮𝑥 − 3 sin﷮𝑥﷯﷯﷮𝑡﷯ ﷯. 𝑑𝑥 = ﷮﷮ 2 cos﷮𝑥 − 3 sin﷮𝑥﷯﷯﷮𝑡﷯ ﷯. 𝑑𝑡﷮4 cos﷮𝑥﷯ − 6 sin﷮𝑥﷯﷯ = ﷮﷮ 2 cos﷮𝑥 − 3 sin﷮𝑥﷯﷯﷯﷮𝑡﷯ ﷯. 𝑑𝑡﷮2 2 cos﷮𝑥﷯ − 3 sin﷮𝑥﷯﷯﷯ = ﷮﷮ 1﷮2𝑡﷯﷯. 𝑑𝑡 = 1﷮2﷯ ﷮﷮ 𝑑𝑡﷮𝑡﷯﷯ = 1﷮2﷯ log﷮ 𝑡﷯﷯+𝐶1 = 1﷮2﷯ log﷮ 4 sin﷮𝑥﷯+6 cos﷮𝑥﷯﷯﷯+𝐶1 = 1﷮2﷯ log﷮ 2 2 sin﷮𝑥﷯+3 cos﷮𝑥﷯﷯﷯﷯+𝐶1 = 1﷮2﷯ log 2+𝑙𝑜𝑔 2 sin﷮𝑥﷯+3 cos﷮𝑥﷯﷯﷯+𝐶1 = 1﷮2﷯ log 2+ 1﷮2﷯ 𝑙𝑜𝑔 2 sin﷮𝑥﷯+3 cos﷮𝑥﷯﷯+𝐶1 = 𝟏﷮𝟐﷯ 𝒍𝒐𝒈 𝟐 𝒔𝒊𝒏﷮𝒙﷯+𝟑 𝒄𝒐𝒔﷮𝒙﷯﷯+𝑪

Ex 7.2, 24 - Chapter 7 Class 12 Integrals (Term 2)

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