Integral of 2x*e^x^2-1 dx

1. Integrate term-by-term:

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

      $$\int 2 x e^{x^{2}}\, dx = 2 \int x e^{x^{2}}\, dx$$

      1. Let $u = e^{x^{2}}$.

         Then let $du = 2 x e^{x^{2}} dx$ and substitute $\frac{du}{2}$:

         $$\int \frac{1}{4}\, du$$

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

            $$\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$$

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

               $$\int 1\, du = u$$

            So, the result is: $\frac{u}{2}$

         Now substitute $u$ back in:

         $$\frac{e^{x^{2}}}{2}$$

      So, the result is: $e^{x^{2}}$

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

      $$\int \left(\left(-1\right) 1\right)\, dx = - x$$

   The result is: $- x + e^{x^{2}}$

2. Add the constant of integration:

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

The answer is:

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