If I₁= ₀^ / 2 x x d x , and I₂= ₀^1 ⁻¹ x x d x , then I₁: I₂ is

Step-by-step solution

Given, I₁= ₀^ / 2 x x d x I₂= ₀^1 ⁻¹ x x d x= ₀^ / 4 t t ^2 t d t Put, ⁻¹ x=t aligned & x= t & d x= ^2 t d t & U . L 4 aligned L. L 0 aligned & = ₀^ / 4 t t t 1 ^2 t d t= ₀^ / 4 t t t d t & = ₀^ / 4 2 t 2 t t d t= ₀^ / 4 2 t 2 t d t aligned aligned & Now, put 2 t=x & 2 d t=d x & d t= d x 2 & U . L 2 & aligned aligned & L . L 0 & =2 ₀^ / 2 x x d x & I₂=2 I₁ & I₁: I₂=2: 1 aligned Hence, option (2) is correct.