int x(5^(-x^2))dx Find the indefinite integral

Indefinite integral are written in the form of int f(x) dx = F(x) +C
 where: f(x) as the integrand
          F(x) as the anti-derivative function 
          C  as the arbitrary constant known as constant of integration
For the given problem int x(5^(-x^2)) dx has an integrand in a form of exponential function.
 To evaluate this, we may let:
u = -x^2 then   du= -2x dx or (-1/2)(du)= x dx .
Applying u-substitution, we get:
int x(5^(-x^2)) dx =int (5^(-x^2)) * x dx
                            =int (5^(u)) *(-1/2du)
                            =-1/2int (5^(u) du)
The integral part resembles the basic integration formula:
int a^u du = a^u/(ln(a))+C
Applying it to the problem: 
-1/2int (5^(u) du)=-1/2 * 5^(u)/ln(5) +C
Pug-in u =-x^2 , we get the definite integral:
-1/2 * 5^(-x^2)/ln(5) +C
 or
- 5^(-x^2)/(2ln(5)) +C