Concept:Some useful formulas are:\(\rm ? x^n dx = \frac {(x^{n+1})} {(n+1)} +C; \ n?1\)\(\rm \int sinax= -\frac{cosax}{a} +C\)Calculation:\(\rm \int(4x^3+4sin2x+\sqrt x)dx\)= \(\)\(\rm \int4x^3dx+\int4sin2x\ dx+\int\sqrt xdx\)= \(\rm 4\times\frac{x^{(3+1)}}{3+1} -4\frac{cos2x}{2}+\frac{x^{(\frac{1}{2}+1)}}{\frac{1}{2}+1} + C\)where c is the constant of integration= \(\rm 4\times\frac{x^4}{4}-4\frac{cos2x}{2}+\frac{2}{3}x^\frac{3}{2}\) + C, = \(\rm x^4 - 2cos2x + \frac{2}{3}x\sqrt{x} +C\)