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

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$\left(x + 4\right) e^{x} = x e^{x} + 4 e^{x}$$

   2. Integrate term-by-term:

      1. Use integration by parts:

         $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

         Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$.

         Then $\operatorname{du}{\left(x \right)} = 1$.

         To find $v{\left(x \right)}$:

         1. The integral of the exponential function is itself.

            $$\int e^{x}\, dx = e^{x}$$

         Now evaluate the sub-integral.

      2. The integral of the exponential function is itself.

         $$\int e^{x}\, dx = e^{x}$$

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

         $$\int 4 e^{x}\, dx = 4 \int e^{x}\, dx$$

         1. The integral of the exponential function is itself.

            $$\int e^{x}\, dx = e^{x}$$

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

      The result is: $x e^{x} + 3 e^{x}$

   Method #2

   1. Use integration by parts:

      $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

      Let $u{\left(x \right)} = x + 4$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$.

      Then $\operatorname{du}{\left(x \right)} = 1$.

      To find $v{\left(x \right)}$:

      1. The integral of the exponential function is itself.

         $$\int e^{x}\, dx = e^{x}$$

      Now evaluate the sub-integral.

   2. The integral of the exponential function is itself.

      $$\int e^{x}\, dx = e^{x}$$

2. Now simplify:

   $$\left(x + 3\right) e^{x}$$

3. Add the constant of integration:

   $$\left(x + 3\right) e^{x}+ \mathrm{constant}$$

The answer is:

$$\left(x + 3\right) e^{x}+ \mathrm{constant}$$