If f(x)=x+22x+3. then ∫f(x)x21/2dxis equal to 12g1+2f(x)1−2f(x)−23h3f(x)+23f(x)−2 +C where

Putting y2=f(x)=x+22x+3, we have x=3y2−21−2y2 and  dx=−2y1−2y22dy.So =−∫y⋅2y1−2y22⋅1−2y23y2−2dy=2∫y22y2−13y2−2dy=−2∫12y2−1−23y2−2dy=12log⁡1+2y1−2y−23log⁡3y+23y−2+CThus g(x)=log⁡|x| and h(x)=log⁡|x|.