Integral of 3x*y-2t dt

1. Integrate term-by-term:

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

      $$\int \left(- 2 t\right)\, dt = - 2 \int t\, dt$$

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

         $$\int t\, dt = \frac{t^{2}}{2}$$

      So, the result is: $- t^{2}$

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

      $$\int 3 x y\, dt = 3 t x y$$

   The result is: $- t^{2} + 3 t x y$

2. Now simplify:

   $$t \left(- t + 3 x y\right)$$

3. Add the constant of integration:

   $$t \left(- t + 3 x y\right)+ \mathrm{constant}$$

The answer is:

$$t \left(- t + 3 x y\right)+ \mathrm{constant}$$