NTA Abhyas JEE Main2020MathematicsDefinite IntegrationPractice
If ∫ 0 ∞ sin ⁡ x x d x = k , then the value of ∫ 0 ∞ s i n 3 x x d x is equal to
Options
- Ak
- Bk 2
- Ck 4
- D2 k
Correct answer
B. k 2
Step-by-step solution
sin ⁡ 3 x = 3 sin ⁡ x - 4 s i n 3 x sin ⁡ 3 x = 3 4 sin ⁡ x - 1 4 sin ⁡ 3 x Now, ∫ 0 ∞ s i n 3 x x d x = 3 4 ∫ 0 ∞ sin ⁡ x x d x - 1 4 ∫ 0 ∞ sin ⁡ 3 x x d x = 3 k 4 - Ι 4 (let) Ι = ∫ 0 ∞ sin ⁡ 3 x x d x Let 3 x = t d x = d t 3 Ι = ∫ 0 ∞ sin ⁡ t t 3 d x 3 = k Required value = 3 k 4 - k 4 = k 2