NTA Abhyas JEE Main2020MathematicsDefinite IntegrationPractice

The value of ∫ 0 π 3 l o g 1 + 3 tan ⁡ x d x is equal to

Options

Aπ log ⁡ 2
Bπ 2 log ⁡ 2
Cπ 3 log ⁡ 2
Dπ 4 log ⁡ 2

Correct answer

C. π 3 log ⁡ 2

Step-by-step solution

Let, Ι = ∫ 0 π 3 l o g 1 + 3 tan ⁡ x d x Ι = ∫ 0 π 3 l o g 1 + 3 t a n π 3 - x d x = ∫ 0 π 3 l o g 1 + 3 t a n 3 - tan ⁡ x 1 + 3 tan ⁡ x d x = ∫ 0 π 3 l o g 1 + 3 tan ⁡ x + 3 - 3 tan ⁡ x 1 + 3 tan ⁡ x d x Ι = ∫ 0 π 3 log ⁡ 4 - l o g 1 + 3 tan ⁡ x d x Ι = log ⁡ 4 π 3 - Ι Ι = π 6 l o g 4 = π 3 l o g 2

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