d x 3-2 2 x = ⁻¹( f (x)) 5 + c , (where c is constant of integration), then f( / 4) has the value

Step-by-step solution

aligned & Let I = d x 3-2 2 x & Put x= t & x= ⁻¹ t aligned aligned & d x= dt 1+ t ^2 & aligned & 2 x= 1- t ^2 1+ t ^2 I & = dt 1+ t ^2 3-2 ( 1- t ^2 1+ t ^2 ) & = dt 3 (1+ t ^2 )-2 (1- t ^2 ) & = dt 3+3 t ^2-2+2 t ^2 & = dt ( 5 t )^2+(1)^2 & = 1 5 ⁻¹( 5 t )+ c aligned aligned I= 1 5 ⁻¹ 5 x+c Comparing with ⁻¹ f (x) 5 , we get aligned & f (x)= 5 x & f ( 4 )= 5 4 = 5 aligned