d x e ^x-1 =2 ⁻¹( f (x))+ c where x 0 and c is a constant of integration, then f (x) is

Step-by-step solution

Let I= d x e ^x-1 Let e ^x-1 = t array ll & e ^x-1= t ^2 & e ^x= t ^2+1 & e ^x ~d x=2 t dt & ~d x= 2 t e ^x dt = 2 t t ^2+1 dt aligned & I aligned = 1 t 2 t t ^2+1 dt & =2 1 t ^2+1 dt & =2 ⁻¹( t )+ c & =2 ⁻¹ ( e ^x-1 )+ c & f (x)= e ^x-1 array [ from (i)]