Solve: \( \int e^{2 x^{2}}(2 x) d x \)

To solve the integral \( \int e^{2x^{2}}(2x) \, dx \), we can use the substitution method. Let \( u = e^{2x^{2}} \). Then, the derivative \( du = 4x e^{2x^{2}} \, dx \) implies that \( \frac{1}{4} du = x e^{2x^{2}} \, dx \). This is a crucial step that shows we need to rearrange our integral slightly. Rephrasing it with this substitution: \[ \int e^{2x^{2}}(2x) \, dx = \int 2x e^{2x^{2}} \, dx \] Now substituting \( du = 4x e^{2x^{2}} \,dx\), we get \[ \int \frac{1}{2} du = \frac{1}{2} u + C = \frac{1}{2} e^{2x^{2}} + C \] So, the integral evaluates to: \[ \int e^{2x^{2}}(2x) \, dx = \frac{1}{2} e^{2x^{2}} + C \] That's your solution!