If d x 1+3 ^2 x = 1 2 ⁻¹( f (x))+ c , where c is -a constant of integration, then f (x) is equal to

Step-by-step solution

aligned & d x 1+3 ^2 x = d x ^2 x+ ^2 x+3 ^2 x & = d x 4 ^2 x+ ^2 x & = ^2 x ~d x 4 ^2 x+1 & = 1 4 ^2 x ~d x ^2 x+ 1 4 & Put t = x dt = ^2 x ~d x & ~d x 1+3 ^2 x = 1 4 dt t ^2+ ( 1 2 )^2 = 1 4 2 ⁻¹(2 t )+ c & = 1 2 ⁻¹(2 t )+ c & = 1 2 ⁻¹(2 x)+ c & f (x)=2 x aligned