JEE Advanced2020MathematicsDefinite IntegrationActual
Which of the following inequalities is/are TRUE?
Options
- A∫ 0 1 x cos x d x ≥ 3 8
- B∫ 0 1 x sin x d x ≥ 3 10
- C∫ 0 1 x 2 cos x d x ≥ 1 2
- D∫ 0 1 x 2 sin x d x ≥ 2 9
Correct answer
A. ∫ 0 1 x cos x d x ≥ 3 8
Step-by-step solution
We know that cos x = 1 − x 2 2 ! + x 4 4 ! - x 6 6 ! + … … … … ⇒ cos x ≥ 1 − x 2 2 ∴ ∫ 0 1 x cos x d x ≥ ∫ 0 1 x 1 − x 2 2 d x = x 2 2 − x 4 8 0 1 = 1 2 − 1 8 = 3 8 So, option A is correct. Now, we also know that sin x = x − x 3 3 ! + x 5 5 ! - x 7 7 ! + … … … … … ⇒ sin x ≥ x − x 3 3 ! ,   x ∈ 0 ,   1 ∴ ∫ 0 1 x sin x d x ≥ ∫ 0 1 x x &