KVPY2019MathematicsDefinite Integration
Let J = ∫ 0 1 x 1 + x 8 d x Consider the following assertions: I . J > 1 4 II . J < π 8 Then,
Options
- Aonly I   is true
- Bonly II is true
- Cboth I and II are true
- Dneither I nor II is true
Correct answer
A. only I   is true
Step-by-step solution
As we know, 1 + x 8 < 2 , ∀ x ∈ 0 , 1 ⇒ 1 2 < 1 1 + x 8 , ∀ x ∈ 0 , 1 ⇒ x 2 < x 1 + x 8 , ∀ x ∈ 0 , 1 ⇒ ∫ 0 1 x 2 d x < ∫ 0 1 x 1 + x 8 d x ⇒ ∫ 0 1 x 1 + x 8 d x > 1 4 ⇒ J > 1 4 Similarly, as we know, 1 + x 8 < 1 + x 4 , ∀ x ∈ 0 , 1 ⇒ ∫ 0 1 x 1 + x 4 d x < ∫ 0 1 x 1 + x 8 d x ⇒ J > 1 2 tan - 1 x 2 0 1 ⇒ J > π 8 Statement J = ∫ 0 1 x 1 + x 8