Question

Let I_n=∫(lnx)n x∈[1,e] dx, n∈N

Statement-1 : I₁ ,I₂ , I₃ . . . is an increasing sequence

Statement-2 : l_n x is an increasing function.

(a) Statement-1 is false, Statement-2 is true. 

(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. 

(d) Statement-1 is true, Statement-2 is false.

Answer 1

Correct option (a)

Explanation:

 Statement – II is true, as if f (x) = ln x, then f'(x)=1/x>0 (as x > 0, so that f (x) is defined) 

Statement – I is not true as 0 < ln x < 1, ∀ x ∈(1, e) and hence (ln x)ⁿ decreases as n is increasing. So that I_n is a decreasing sequence