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NTA Abhyas JEE Main2020MathematicsIndefinite IntegrationPractice

The indefinite integral I = ∫ sin 2 ⁡ x - cos 2 ⁡ x 2019 sin ⁡ x 2021 cos ⁡ x 2021 d x simplifies to (where c is an integration constant)

Options

  1. Asin 2 ⁡ x - cos 2 ⁡ x 2020 2020 + c
  2. Btan ⁡ x - cot ⁡ x 2020 2020 + c
  3. Csin ⁡ x - cos ⁡ x 2020 2020 + c
  4. Dtan 2 ⁡ x + cot 2 ⁡ x 2020 2020 + c

Correct answer

B. tan ⁡ x - cot ⁡ x 2020 2020 + c

Step-by-step solution

I = ∫ sin 2 ⁡ x - cos 2 ⁡ x sin ⁡ x cos ⁡ x 2019 1 sin 2 ⁡ x cos 2 ⁡ x d x = ∫ tan ⁡ x - cot ⁡ x 2019 sin 2 ⁡ x + cos 2 ⁡ x sin 2 ⁡ x cos 2 ⁡ x d x = ∫ tan ⁡ x - cot ⁡ x 2019 sec 2 ⁡ x + c o s e c 2 x d x Let tan ⁡ x - cot ⁡ x = t ⇒ sec 2 ⁡ x + cosec 2 x d x = d t ∴ I = ∫ t 2019 d t = t 2020 2020 + c = tan ⁡ x - cot ⁡ x 2020 2020 + c

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