Integral of (x^3+2x)sinh(x) dx

1. Rewrite the integrand:

   $$\left(x^{3} + 2 x\right) \sinh{\left(x \right)} = x^{3} \sinh{\left(x \right)} + 2 x \sinh{\left(x \right)}$$

2. Integrate term-by-term:

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

      But the integral is

      $$x^{3} \cosh{\left(x \right)} - 3 x^{2} \sinh{\left(x \right)} + 6 x \cosh{\left(x \right)} - 6 \sinh{\left(x \right)}$$

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

      $$\int 2 x \sinh{\left(x \right)}\, dx = 2 \int x \sinh{\left(x \right)}\, dx$$

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

         But the integral is

         $$x \cosh{\left(x \right)} - \sinh{\left(x \right)}$$

      So, the result is: $2 x \cosh{\left(x \right)} - 2 \sinh{\left(x \right)}$

   The result is: $x^{3} \cosh{\left(x \right)} - 3 x^{2} \sinh{\left(x \right)} + 8 x \cosh{\left(x \right)} - 8 \sinh{\left(x \right)}$

3. Add the constant of integration:

   $$x^{3} \cosh{\left(x \right)} - 3 x^{2} \sinh{\left(x \right)} + 8 x \cosh{\left(x \right)} - 8 \sinh{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$x^{3} \cosh{\left(x \right)} - 3 x^{2} \sinh{\left(x \right)} + 8 x \cosh{\left(x \right)} - 8 \sinh{\left(x \right)}+ \mathrm{constant}$$