Ex 7.2, 5 - Integrate sin (ax + b) cos (ax+b) - Ex 7.2

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.2, 5 (Method 1) Integrate the function: sin﷮ 𝑎𝑥+𝑏﷯﷯ cos﷮ 𝑎𝑥+𝑏﷯﷯ Step 1: Let sin﷮ 𝑎𝑥+𝑏﷯﷯=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 cos⁡(𝑎𝑥 + 𝑏) . 𝑑 𝑎𝑥+ 𝑏﷯﷮𝑑𝑥﷯ = 𝑑𝑡﷮𝑑𝑥﷯ cos⁡(𝑎𝑥 + 𝑏) 𝑎+0﷯= 𝑑𝑡﷮𝑑𝑥﷯ cos⁡(𝑎𝑥 + 𝑏) . 𝑎= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥 = 𝑑𝑡﷮𝑎 . cos﷮ 𝑎𝑥+𝑏﷯﷯﷯ Step 2: Integrating the function ﷮﷮ sin⁡(𝑎𝑥+𝑏) cos⁡(𝑎𝑥+𝑏)﷯. 𝑑𝑥 Putting 𝑡= 𝑠𝑖𝑛﷮ 𝑎𝑥+𝑏﷯﷯ & dx = 𝑑𝑡﷮𝑎 . cos﷮ 𝑎𝑥+𝑏﷯﷯﷯ = ﷮﷮ 𝑡﷯. cos﷮ 𝑎𝑥+𝑏﷯﷯ . 𝑑𝑡﷮𝑎 . cos﷮ 𝑎𝑥+𝑏﷯﷯﷯ = 1﷮𝑎﷯ ﷮﷮ 𝑡﷯. 𝑑𝑡 = 1﷮𝑎﷯ 𝑡﷮1+1﷯﷮1 +1﷯ +𝐶1﷯ = 1﷮𝑎﷯ 𝑡﷮2﷯﷮2﷯ +𝐶1﷯ = 𝑡﷮2﷯﷮2𝑎﷯ + 𝐶1﷮𝑎﷯ = 𝑡﷮2﷯﷮2𝑎﷯ + 𝐶 = sin﷮2﷯﷮ 𝑎𝑥+𝑏﷯﷯﷮2𝑎﷯ +𝐶 We know that cos﷮2𝑥﷯=1−2 sin﷮2﷯﷮𝑥﷯ ⇒ cos﷮2﷯ 𝑎𝑥+𝑏﷯=1−2 sin﷮2﷯﷮ 𝑎𝑥+𝑏﷯﷯ ⇒2 sin﷮2﷯﷮ 𝑎𝑥+𝑏﷯﷯=1− cos﷮2﷯ 𝑎𝑥+𝑏﷯ ⇒ sin﷮2﷯﷮ 𝑎𝑥+𝑏﷯﷯= 1﷮2﷯− 1﷮2﷯ cos﷮2﷯ 𝑎𝑥+𝑏﷯ Putting in (1) = 1﷮2 × 2𝑎﷯ − 1﷮2 × 2𝑎﷯ cos﷮2﷯ 𝑎𝑥+𝑏﷯+𝐶 = −𝟏﷮𝟒𝒂﷯ 𝒄𝒐𝒔﷮𝟐﷯ 𝒂𝒙+𝒃﷯+𝑪𝟏 Ex 7.2, 5 (Method 2) Integrate the function: sin⁡(𝑎𝑥 + 𝑏) cos⁡(𝑎𝑥 + 𝑏) Step 1: Taking the given function sin﷮ 𝑎𝑥 + 𝑏﷯﷯ cos﷮ 𝑎𝑥 + 𝑏﷯﷯ = 1﷮2﷯ sin﷮ 2 𝑎𝑥+𝑏﷯﷯﷯ = 1﷮2﷯ sin﷮ 2𝑎𝑥+2𝑏﷯﷯ Step 2: Let 2𝑎𝑥+2𝑏=𝑡 Let 2𝑎𝑥+2𝑏=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 2𝑎+0= 𝑑𝑡﷮𝑑𝑥﷯ 2𝑎= 𝑑𝑡﷮𝑑𝑥﷯ 2𝑎.𝑑𝑥=𝑑𝑡 𝑑𝑥= 𝑑𝑡﷮2𝑎﷯ Step 3: Integrating the function ﷮﷮ sin⁡(𝑎𝑥+𝑏) cos⁡(𝑎𝑥+𝑏) ﷯ .𝑑𝑥 = 1﷮2﷯ ﷮﷮sin⁡(2𝑎𝑥+2𝑏)﷯ .𝑑𝑥 Putting 𝑡=2𝑎𝑥+2𝑏 & 𝑑𝑥= 𝑑𝑡﷮2𝑎﷯ = 1﷮2﷯ ﷮﷮sin⁡(𝑡)﷯ . 𝑑𝑡﷮2𝑎﷯ = 1﷮4𝑎﷯ ﷮﷮sin⁡(𝑡)﷯ .𝑑𝑡 = 1﷮4𝑎﷯ − cos﷮𝑡﷯+𝐶1﷯ = − 1﷮4𝑎﷯ . cos﷮𝑡﷯ + 𝐶1﷮4𝑎﷯ = − 1﷮4𝑎﷯ . cos﷮𝑡﷯ + 𝐶 = − 1﷮4𝑎﷯ . cos﷮(2𝑎𝑥+2𝑏)﷯ + 𝐶 = − 𝟏﷮𝟒𝒂﷯ . 𝒄𝒐𝒔﷮𝟐(𝒂𝒙+𝒃)﷯+𝑪

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