Let f(x) be a positive function and I₁= _ - 1 2 ^1 2 x f(2 x(1-2 x)) d x and I₂= _ -1 ^2 f(x(1-x)) d x . Then the value of I₂ I₁ is equal to ________

Step-by-step solution

aligned & I ₁= _ - 1 2 ^1 2 xf (2 x (1-2 x )) dx & 2 x = t 2 dx = dt I ₁= 1 2 _ -1 ^2 tf ( t (1- t )) dt & 2 I ₁= _ -1 ^2(1- t ) f (1- t )(1-(1- t )) dt & 2 I ₁= _ -1 ^2 f ( t (1- t ) dt - _ -1 ^2 tf ( t (1- t ) dt . & 2 I ₁= I ₂-2 I ₁ & 4 I ₁= I ₂ & I ₂ I ₁ =4 aligned