`int 2x^3 e^(x^2) dx`

Explanation

`\int 2 x^{3} e^{x^{2}} d x=\int 2 x \cdot x^{2} e^{x^{2}}`
Let, `\quad x^{2}=t `
`\Rightarrow 2 x d x=d t `
`\therefore \int 2 x^{3} e^{x^{2}} d x=\int t \cdot e^{t} d t`
`=t \int e^{t} d t+\int[\frac{d t}{d t} \cdot \int e^{t} d t] d t`
`=t \cdot e^{t}+e^{t}+C `
`=x^{2} e^{x^{2}}+e^{x^{2}}+C`