Integral of 4+sqrt(x)-e^(4*x) dx

1. Integrate term-by-term:

   1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

      $$\int \sqrt{x}\, dx = \frac{2 x^{\frac{3}{2}}}{3}$$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- e^{4 x}\right)\, dx = - \int e^{4 x}\, dx$$

      1. Don't know the steps in finding this integral.

         But the integral is

         $$\frac{e^{4 x}}{4}$$

      So, the result is: $- \frac{e^{4 x}}{4}$

   1. The integral of a constant is the constant times the variable of integration:

      $$\int 4\, dx = 4 x$$

   The result is: $\frac{2 x^{\frac{3}{2}}}{3} + 4 x - \frac{e^{4 x}}{4}$

2. Add the constant of integration:

   $$\frac{2 x^{\frac{3}{2}}}{3} + 4 x - \frac{e^{4 x}}{4}+ \mathrm{constant}$$

The answer is:

$$\frac{2 x^{\frac{3}{2}}}{3} + 4 x - \frac{e^{4 x}}{4}+ \mathrm{constant}$$