Example 35  - Evaluate dx / 1 + root tan x - Chapter 7 CBSE - Examples

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 35 (Method 1) Evaluate 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 𝑑𝑥﷮1 + ﷮ tan﷮𝑥﷯﷯﷯﷯ Let I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1﷮1 + ﷮ tan﷮𝑥﷯﷯﷯﷯.𝑑𝑥 I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1﷮1 + ﷮ sin﷮𝑥﷯﷮ cos﷮𝑥﷯﷯﷯﷯﷯.𝑑𝑥 I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1﷮1 + ﷮ sin﷮𝑥﷯﷯﷮ ﷮ cos﷮𝑥﷯﷯﷯﷯﷯.𝑑𝑥 I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1﷮ ﷮ cos﷮𝑥﷯﷯ + ﷮ sin﷮𝑥﷯﷯﷮ ﷮ cos﷮𝑥﷯﷯﷯﷯﷯ 𝑑𝑥 I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ ﷮ cos﷮𝑥﷯﷯ ﷮ ﷮ cos﷮𝑥﷯﷯ + ﷮ sin﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 ∴ I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ ﷮ cos﷮ 𝜋﷮6﷯ + 𝜋﷮3﷯ − 𝑥﷯﷯﷯ ﷮ ﷮ cos ﷮ 𝜋﷮6﷯ + 𝜋﷮3﷯ − 𝑥﷯﷯﷯ + ﷮ sin ﷮ 𝜋﷮6﷯ + 𝜋﷮3﷯ − 𝑥﷯﷯﷯﷯﷯ 𝑑𝑥 I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ ﷮ cos﷮ 𝜋﷮2﷯ − 𝑥﷯﷯﷯ ﷮ ﷮ cos﷮ 𝜋﷮2﷯ − 𝑥﷯﷯﷯ + ﷮ sin﷮ 𝜋﷮2﷯ − 𝑥﷯﷯﷯﷯﷯ 𝑑𝑥 I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ ﷮ sin﷮𝑥﷯﷯ ﷮ ﷮ sin﷮𝑥﷯﷯ + ﷮ cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 Adding (1) and (2) i.e. (1) + (2) I + I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ ﷮cos 𝑥﷯ ﷮ ﷮ sin﷮𝑥﷯﷯ + ﷮ cos﷮𝑥﷯﷯﷯﷯. 𝑑𝑥+ 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ ﷮ sin﷮𝑥﷯﷯ ﷮ ﷮ sin﷮𝑥﷯﷯ + ﷮ cos﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 2I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ ﷮cos 𝑥﷯ + ﷮ sin﷮𝑥﷯﷯﷮ ﷮cos 𝑥﷯ + ﷮ sin﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 2I= 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮1.﷯ 𝑑𝑥 2I= 𝑥﷯﷮ 𝜋﷮6﷯﷮ 𝜋﷮3﷯﷯ I= 1﷮2﷯ 𝜋﷮3﷯− 𝜋﷮6﷯﷯ I= 1﷮2﷯ 2𝜋 − 𝜋﷮6﷯﷯ 𝐈= 𝝅﷮𝟏𝟐﷯ Example 35 (Method 2) Evaluate 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 𝑑𝑥﷮1 + ﷮ tan﷮𝑥﷯﷯﷯﷯ Let I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1﷮1 + ﷮ tan﷮𝑥﷯﷯﷯﷯.𝑑𝑥 I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1﷮1 + ﷮ tan﷮ 𝜋﷮3﷯ + 𝜋﷮6﷯ − 𝑥﷯﷯﷯﷯﷯ 𝑑𝑥 = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1﷮1 + ﷮ tan﷮ 𝜋﷮2﷯ − 𝑥﷯﷯﷯﷯﷯ 𝑑𝑥 I = 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1 ﷮1 + ﷮ co𝑡﷮𝑥﷯﷯ ﷯﷯ 𝑑𝑥 Adding (1) and (2) I+I= 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1 ﷮ 1 + ﷮ tan﷮𝑥﷯﷯﷯+ 1﷮1 + ﷮ cot﷮𝑥﷯﷯﷯﷯﷯ 𝑑𝑥 2I= 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 1 + ﷮ cot﷮𝑥﷯﷯ + 1 + ﷮ tan﷮𝑥﷯﷯ ﷯﷮ 1 + ﷮tan 𝑥﷯ ﷯ 1 + ﷮ cot﷮𝑥﷯﷯ ﷯﷯﷯ 𝑑𝑥 2I= 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 2 + ﷮ cot﷮𝑥﷯﷯ + ﷮ tan﷮𝑥﷯﷯﷮ 1 + ﷮ cot﷮𝑥﷯﷯ + ﷮ tan﷮𝑥﷯﷯ + ﷮ 𝐭𝐚𝐧﷮𝒙 ﷯﷯ . ﷮ 𝐜𝐨𝐭﷮𝒙﷯﷯﷯﷯ 𝑑𝑥 2I= 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 2 + ﷮ cot﷮𝑥﷯﷯ + ﷮ tan﷮𝑥﷯﷯﷮ 1 + ﷮ cot﷮𝑥﷯﷯ + ﷮ tan﷮𝑥﷯﷯ + 𝟏﷯﷯ 𝑑𝑥 2I= 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮ 2 + ﷮ cot﷮𝑥﷯﷯ + ﷮ tan﷮𝑥﷯﷯﷮2 + ﷮ cot﷮𝑥﷯﷯ + ﷮ tan﷮𝑥﷯﷯﷯﷯ 𝑑𝑥 2I= 𝜋﷮6﷯﷮ 𝜋﷮3﷯ ﷮𝑑𝑥﷯ 2I= I﷯﷮ 𝜋﷮6﷯﷮ 𝜋﷮3﷯﷯ I= 1﷮2﷯ 𝜋﷮3﷯− 𝜋﷮6﷯﷯ I= 1﷮2﷯ 𝜋﷮6﷯﷯ 𝐈= 𝝅﷮𝟏𝟐﷯

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