NTA Abhyas JEE Main2020MathematicsIndefinite IntegrationPractice

If I = ∫ tan ⁡ x + cot ⁡ x d x , then I equals to (where, C is an arbitrary constant)

Options

A2 sin - 1 ⁡ sin ⁡ x + cos ⁡ x + C
B2 cos - 1 ⁡ sin ⁡ x - cos ⁡ x + C
C2 sin - 1 ⁡ sin ⁡ x - cos ⁡ x + C
D2 cos - 1 ⁡ sin ⁡ x + cos ⁡ x + C

Correct answer

C. 2 sin - 1 ⁡ sin ⁡ x - cos ⁡ x + C

Step-by-step solution

I = ∫ sin ⁡ x + cos ⁡ x cos ⁡ x sin ⁡ x d x Put sin ⁡ x - cos ⁡ x = t , so that sin ⁡ x + cos ⁡ x d x = d t and 1 – 2 sin ⁡ x cos ⁡ x = t 2 ∴ I = 2 ∫ d t 1 - t 2 = 2 sin - 1 ⁡ t + C = 2 sin - 1 ⁡ sin ⁡ x - cos ⁡ x + C

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